Specific Heat of Sr 3CuPt. 5 1r. 5 0 6 Below 1K
by
Adam D. Poleyn
A.B. Physics
Princeton University (1992)
Submitted to the Department of Physics
in partial fulfillment of the requirements for the degree of
Doctor of Philosophy
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June 1999
© Massachusetts Institute of Technology 1999. All rights reserved.
Author ...........................
......................
Department of Physics
May 6, 1999
Certified by .......... /..-.-.
. 1 ....... . . . . . . . .....
Thomas J. Greytak
Professor of Physics
Thesis Supervisor
Accepted by......... . ...
Asso..
........
....
...
7...f...........
Thomas J. Greytak
Professor, Associate Dep rtment Head for Education
MASSACHUSETT$ NSTITUTE
LIBRARIE
! IES
Specific Heat of Sr 3CuPtO.Oro. 5 0 6 Below 1K
by
Adam D. Polcyn
Submitted to the Department of Physics
on May 6, 1999, in partial fulfillment of the
requirements for the degree of
Doctor of Philosophy
Abstract
The alloy Sr 3 CuPtO.5 Iro.0 O6 was first fabricated by zur Loye and his collaborators at
MIT in 1994. Lee and his collaborators have modeled the material as a spin-' chain
with randomly distributed ferromagnetic (FM) and antiferromagnetic (AF) nearest
neighbor bonds of equal strength. Magnetization measurements indicate that the
material obeys the Curie Law to 4K, even though the AF and FM interactions have
strengths of order 30K. Lee's model explains this unusual Curie Law behavior, and
predicts that at temperatures well below 1K, the material should exhibit a scaling
behavior in specific heat and susceptibility characteristic of a new universality class
of disordered quantum spin systems, and contain an unusually large amount of spin
entropy. If a large amount of spin entropy is available in the material below 1K,
the material could be useful for magnetic refrigeration to temperatures as low as
100pK. Motivated by the work of zur Loye and Lee, I have constructed a new
apparatus to measure specific heat u of Sr 3 CuPtO.5 Iro.50 6 between 0.1K and 1K in
fields to 7T. I have developed thermal characterization techniques to verify that the
thermal properties of the calorimeter are appropriate for specific heat measurement
below 1K, and used the apparatus to measure specific heat of potassium ferricyanide
K 3 Fe(CN) 6 and Sr 3 CuPtO.5 1ro.5 0
6
using the AC and thermal relaxation methods. I
find no evidence for a phase transition to a long-range ordered state between 0.1K
and 2K in Sr 3 CuPtO.5 Iro.0 O 6 , and that u is consistent with the scaling law predicted
by Lee's model between 0.1K and 0.4K at zero field. Application of fields below 10kG
suppresses a, and at 10kG or obeys a T3/ 2 power law below 0.5K. I show that given
my data, Sr 3 CuPtO. 5 Iro. 5 0 6 is not superior to known paramagnetic salts for magnetic
refrigeration to lmK, and suggest that a successful understanding of the physics of
Sr 3 CuPtO.5 IrO. 5 0 6 below 1K may require consideration of interactions between spins
on different chains.
Thesis Supervisor: Thomas J. Greytak
Title: Professor of Physics
Acknowledgements
The work described in this thesis is a testimony to the love, support, and patience of
my wife, Amy Fronduti Polcyn. She has been a wonderful friend and helper to me
throughout my years as a graduate student, and I am grateful and honored to be her
husband.
I would also like to thank my advisor, Professor Thomas Greytak, for his support
and patience. The laboratory environment that he has created is an excellent one in
which to obtain an education in experimental physics, and I am honored to have been
his student and to have worked in this laboratory. Despite the inevitable frustrations
and setbacks, I have very much enjoyed working on this project and learning about
the physics of magnetic materials, and am grateful to him for proposing this project
for my thesis work.
I have been blessed with an excellent group of colleagues during my graduate
career. I have learned a great deal from each of them, and wish I had taken even
more advantage of the pool of talent that exists in this laboratory. Many of the
key ideas in this thesis resulted from conversations with them. In addition to their
talents as scientists, they are an excellent group of people, always ready to listen,
teach, help, and support. My classmate Dale Fried has been a great friend, brother,
and confidant to me. Tom Killian was an excellent office mate and colleague, and
probably took as many phone messages for me as I did for him. I have had many
enjoyable conversations with Lorenz Willmann. I am grateful to Stephen Moss for
his friendship and support, and for many enjoyable hours on the tennis court. It has
been a pleasure to work with David Landhuis, and I am grateful to him for taking
over as safety officer. In addition to these current members of the group, I would like
to acknowledge past members of the group, from whom I have learned most of what I
know: Mike Yoo, Claudio Cesar, Albert Yu, Jon Sandberg, and John Doyle. Finally,
I would like to acknowledge Professor Daniel Kleppner, with whom I worked early in
my graduate career. It was always a pleasure to talk with him about physics and to
listen to his stories.
I have also had the pleasure of supervising several undergraduate projects during
my graduate career. It was a pleasure to supervise Carlo Mattoni's thesis, and a lot
of fun to work with him. I have also enjoyed working with Mihai Ibanescu, who I in
many ways consider more of a colleague than a student.
Paul Jackson has been an excellent friend and brother to me over the latter half of
my years at MIT. His constant support, prayers, and concern have been a great source
of encouragement to me, as have the many joyful times we have shared together.
There are many other people who have made my years at MIT more enjoyable,
and supported me along the way. Among them, I would like to mention Bryan Atchison, Jeff Niemann, Scott Socolofsky, members of the Eastgate Graduate Christian
Fellowship Bible Study, and members of my St. Ignatius small group, especially Ed
and Mary Dailey.
I would also like to acknowledge the love and support of other members of my
family, especially my parents, Dr. Daniel Polcyn and Elizabeth Polcyn. They have
been a constant source of encouragement, comfort, and support. I am grateful to my
son, Stephen, who always smiled for me when I came home after a long day at the
lab and has brought both Amy and me so much joy. I would also like to acknowledge
the support and love of Anne and Bob Fronduti, John and Meghan Fronduti, Karen
Fronduti, and Sarah Polcyn.
Finally, I would like to acknowledge and thank my Lord and Savior, Jesus Christ.
He is the one who has put all these people in my life, Who has given me the opportunity to study physics, and Who has created the amazing things I have had the
pleasure to study as a graduate student.
Psalm 116
I love the Lord, because He hears
My voice and my supplications.
Because He has inclined His ear to me,
Therefore I shall call upon Him as long as I live.
The cords of death encompassed me,
And the terrors of Sheol came upon me;
I found distress and sorrow.
Then I called upon the name of the Lord:
"0 Lord, I beseech Thee, save my life!"
Gracious is the Lord, and righteous;
Yes, our God is compassionate.
The Lord preserves the simple;
I was brought low, and He saved me.
Return to your rest, 0 my soul,
For the Lord has dealt bountifully with you.
For Thou hast rescued my soul from death,
My eyes from tears,
My feet from stumbling.
I shall walk before the Lord
In the land of the living.
I believed when I said,
"I am greatly afflicted."
I said in my alarm,
"All men are liars."
What shall I render to the Lord
For all His benefits toward me?
I shall lift up the cup of salvation,
And call upon the name of the Lord,
Oh may it be in the presence of all His people.
Precious in the sight of the Lord
Is the death of His godly ones.
O Lord, surely I am Thy servant,
I am Thy servant, the son of Thy handmaid,
Thou hast loosed my bonds.
To Thee I shall offer a sacrifice of thanksgiving,
And call upon the name of the Lord.
I shall pay my vows to the Lord,
Oh may it be in the presence of all His people,
In the courts of the Lord's house,
In the midst of you, 0 Jerusalem.
Praise the Lord!
For Amy
Contents
1
Introduction
1.1 Properties of Spin Chains
1.2
1.3
2
3
. . . . . . . . . . . . . . . . . . . . . . . .
21
23
Sr 3 CuPt1pIrpO6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Specific Heat of Sr 3 CuPtO.5 1ro. 5 0 6 below 1K . . . . . . . . . . . . . .
32
26
Random Quantum Spin Chains and Sr 3 CuPt 1 _pIrpO 6
35
2.1
Theoretical Work on RQSC
2.2
Experimental Work on Sr 3 CuPt1pIrpO 6
. . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . .
35
43
Methods
3.1 Quasi-Adiabatic Calorimetry . . . . . . . . . . . . . . . . . . . . . . .
3.2 Thermal Relaxation Calorimetry . . . . . . . . . . . . . . . . . . . .
3.3 AC Calorim etry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
51
51
52
54
4 Apparatus
4.1 Dilution Refrigerator and Magnet
4.2 Heat Capacity Experiment . . . .
4.2.1 Support Structure . . . . .
4.2.2 Calorim eter . . . . . . . .
4.2.3 Calorimeter Models . . . .
4.2.4 Thermometry . . . . . . .
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59
59
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63
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67
5 Potassium Ferricyanide Experiment
5.1 Calorimeter Preparation . . . . . . . . .
5.2 AC Method Procedures . . . . . . . . . .
5.3 Thermal Relaxation Method Procedures
5.4 Empty Calorimeter Results . . . . . . .
5.5 K 3Fe(CN) 6 Calorimeter Results . . . . .
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69
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87
87
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93
93
6
Sr 3 CuPtO.5 Iro. 5 0 6 Experiment
6.1 Calorimeter Preparation . .
6.2 AC Method Procedures . . .
6.3 Thermal Relaxation Method
6.4 Empty Calorimeter Results
6.4.1 AC Method Results .
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Procedures
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11
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6.5
6.6
7
6.4.2 Relaxation Method Results .
Sr 3CuPtO.5 IrO. 5 0 6 Calorimeter Results
6.5.1 AC Method Results . . . . . .
6.5.2 Relaxation Method Results .
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96
103
103
107
111
Sr 3 CuPtO.5 IrO. 5 0 6 Specific Heat Determination
Conclusions and Future Work
7.1
Sr 3 CuPtO.5 IrO. 5 0 6 Specific Heat Results . . . . . . . . . . . . . . . . .
7.2
7.3
7.4
7.5
Discussion
Discussion
Discussion
Discussion
7.6
7.7
I: Entropy, Field Dependence . . . . . . . . . .
II: Comparison with RQSC Theory . . . . . . .
III: Comparison with Beauchamp Results . . .
IV: Miscellaneous Interpretations . . . . . . . .
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Prospects for Adiabatic Demagnetization of Sr 3 CuPtO.5 Iro. 5 0 6 .
Future W ork . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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113
113
116
119
120
121
122
124
A Calorimeter Conductance from Power/Temperature Curves
127
B Exponential Fits with Instrumental Response
133
C Schwall Model
135
D Model for AC Transfer Function in Presence of - 2 Effect
139
Bibliography
143
12
List of Figures
1-1
1-2
1-3
1-4
1-5
Ordered Ising chain at T = 0 (top) and Ising chain with fluctuation
(bottom ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Ordered Ising net at T = 0 (left) and Ising net with fluctuation (right).
Specific heat of various types of spin chains. The AF and FM chain
results were taken from [5], extrapolated to zero temperature using
simple spin wave theory, and the classical result from [4]. These results
are plotted assuming the Furusaki Hamiltonian (1.2); the Bonner and
Fisher Hamiltonian uses 2J rather than J. . . . . . . . . . . . . . . .
Representation of T = 0 FM spin wave. Rather than reducing the
chain magnetization by breaking the chain at a single point, which
costs energy 2J, the magnetization is reduced in a spin wave by tilting
all spins slightly off the z axis, which costs much less energy. The spin
wave is a traveling wave in which the difference in azimuthal angle
between adjacent spin sites in the wave is constant. . . . . . . . . . .
Crystal structure of Sr 3 MM'0
6
22
22
24
25
. Left: view along the length of a chain,
showing alternating MO 6 trigonal prisms (M represented by the small
ball at center of prism) and M'O 6 octahedra (M' represented by the
large ball at center of octahedron). Right: view down the c axis.
Clusters of 3 Sr ions surround each chain . . . . . . . . . . . . . . . .
27
M/H data for (top left) Sr 3 CuPtO6 , (top right) Sr 3 CuIrO6 , and
Sr 3 CuPt0 .5 Ir 0 .5 0 6 (bottom). . . . . . . . . . . . . . . . . . . . . . . .
29
1-7
1/x data for Sr 3 CuPtI_,IrO
. . . . . . . . .
30
1-8
1/X theoretical prediction of Lee and collaborators [17]. . . . . . . . .
31
2-1
Schematic view of spin wave confinement. Due to different dispersion
relations for FM and AF segments, spin waves from a given segment
do not propagate into adjacent segments. A mechanical analogy is a
rope with discontinous changes in thickness. . . . . . . . . . . . . . .
Missing entropy as a function of p, calculated with (line) Equation 2.10
and (open circles) HTE. . . . . . . . . . . . . . . . . . . . . . . . . .
Spin chain considered by Westerberg et al. (Equation 2.17). The
FM segments form large effective spins, separated by AF segments of
variable length. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1-6
2-2
2-3
6
taken by Nguyen [15].
13
36
38
39
2-4
Schematic of the RSRG method. A renormalization step proceeds by
first identifying Ao, the strongest bond in the chain. Then the two
spins linked by Ao are frozen to form an effective spin SL + SR, and
the nearest neighbor bonds renormalized to A1 and A 2 . As the chain
Ao, it effectively carries out a step in the RSRG. .
40
M versus H data of Beauchamp and Rosenbaum on p = 0.5, 0.667
sam ples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
45
. . . . . . . .
46
cools below kBT
2-5
-
2-6
AC susceptibility data of Beauchamp and Rosenbaum.
2-7
Recent AC susceptibility data on Sr 3 CuPt. 5 Iro.5O 6 taken by Beauchamp. 47
2-8
Magnetization versus field for Sr 3 CuPtO.5 1r o.5 0 6 samples used in this
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
48
Heat capacity data of Ramirez. In each case, the sample consisted of
0.2g of the Sr 3 MM'0 6 material, and 0.2g of silver powder, which were
compressed together to form a pellet. The pellet was then glued to the
calorim eter [32]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
49
. . . . . . . . . . . . . . . . . . .
53
thesis.
2-9
3-1
Thermal circuit for Schwall model.
3-2
Thermal circuit for Sullivan and Siedel model. The calorimeter and
sample are assumed to be in excellent thermal contact, so that the
slab with heat capacity Ct0, represents the entire sample/calorimeter
assembly. Heat flux = -e"t (a is slab cross-sectional area) is applied
at one end of the calorimeter by the heater, and temperature T =
Td, + Tac is sensed at the other side of the calorimeter. Heat is dumped
to thermal ground (the heat bath) through the weak link conductance
K . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .
4-1
4-2
4-3
4-4
Overall view of apparatus. Only one calorimeter stage is shown. Note
that devices (heater, thermometer) are not shown on the calorimeter.
.
54
61
Wireframe view of calorimeter stage. The calorimeter sandwich (sample dough between two quartz plates) is supported by vertical and
horizontal vespel pegs. These pegs are glued into brass L brackets. . .
62
Top view of one calorimeter plate, showing configuration of heater,
thermometer, and thermal link. . . . . . . . . . . . . . . . . . . . . .
62
Expected heat capacity of various calorimeter components. The N
grease data below 0.4 K was obtained by linear extrapolation, which
is reasonable for glassy materials [47]. The quartz data below 0.3 K is
a T 3 extrapolation, which underestimates the true heat capacity [47].
Also shown are expected empty calorimeter heat capacity and measured empty calorimeter heat capacity from Sr 3 CuPt. 5 Iro. 5O 6 experiment, and Sr 3 CuPtO.5 1r o .5 0 6 heat capacity measured here (below 1K)
and by Ramirez [15](above 1K). . . . . . . . . . . . . . . . . . . . . .
14
65
4-5
Expected thermal conductance of various calorimeter components. The
PtW entry is for 16 1 mil diameter PtW wires, each 12 inch long (representing the heater and thermometer lead wires). The NbTi entry is
for the same number and lengths of 5 mil diameter NbTiwires. The
quartz/sample boundary resistance is taken to be the same as that
between copper and glue. . . . . . . . . . . . . . . . . . . . . . . . . .
66
5-1
Monoclinic unit cell of K3 Fe(CN) 6 . Closed spheres represent Fe, open
spheres K, open diamonds C, and closed squares N. 3, the angle between a and c, is approximately 107', and a = 7.04A, b = 10.44A,
and c = 8.4A. Six cyanide groups surround each Fe, in approximately
octahedral coordination. Here, only two Fe ions are shown with all
cyanide groups. The closest cyanide groups on nearest neighbor sites
are 2.74A apart on the chain axis a. For nearest neighbors along b, the
closest cyanide groups are 6.14A apart. . . . . . . . . . . . . . . . . .
70
5-2
Apparatus used to measure AC heat capacity for Sr 3 CuPtO.5 Iro. 5 0 6 and
5-13
K3 Fe(CN) 6 experiment. . . . . . . . . . . . . . . . . . . . . . . . . . .
LR-400 bridge transfer function measured with JFET . . . . . . . . .
Effect of LR-400 transfer function on empty calorimeter 102 mK thermal transfer function. The transfer function is normalized for power.
Typical power/temperature curve for K3 Fe(CN) 6 calorimeter, without
baseline drift correction. . . . . . . . . . . . . . . . . . . . . . . . . .
Typical power/temperature curve for K3 Fe(CN) 6 calorimeter, with
baseline drift correction. . . . . . . . . . . . . . . . . . . . . . . . . .
Empty calorimeter thermal transfer functions, measured and as calculated using the two-wire model and published data on calorimeter
m aterials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Upper limit on AC empty calorimeter heat capacity. . . . . . . . . . .
K3 Fe(CN) 6 calorimeter 108 mK thermal transfer function, with fit to
two-wire model and resulting SSGI transfer function. . . . . . . . . .
K3 Fe(CN) 6 calorimeter 140 mK thermal transfer function, with fit to
two-wire model and resulting SSGI transfer function. . . . . . . . . .
K3 Fe(CN) 6 calorimeter 303 mK thermal transfer function, with fit to
two-wire model and resulting SSGI transfer function. . . . . . . . . .
K3 Fe(CN) 6 calorimeter zero field AC and relaxation heat capacities,
with data for K3 Fe(CN) 6 single crystals published by Fritz. . . . . . .
K3 Fe(CN) 6 calorimeter field-dependent AC heat capacity. . . . . . . .
6-1
Typical power/temperature curve for the Sr 3 CuPtO.5 1ro. 5 0 6 experi-
5-3
5-4
5-5
5-6
5-7
5-8
5-9
5-10
5-11
5-12
6-2
6-3
6-4
ment, with drift correction.. . . . . . . . . . . . . . . . . . . . .
Apparatus used to measure relaxation heat capacity
Sr 3 CuPtO.5 1ro.5 0 6 experiment. . . . . . . . . . . . . . . . . . . .
Raw AT(t) (bottom curve) and AT(t) after averaging ten decays
curve). Top curve is shifted up by 2 mK for comparison. . . . .
Empty calorimeter 128 mK thermal transfer functions. . . . . .
15
. . .
for
. . .
(top
. . .
. . .
72
74
75
76
77
78
79
80
80
81
83
84
90
91
92
93
94
Empty calorimeter 400 mK thermal transfer functions. . . . . . . . .
and
correction,
Empty calorimeter AC heat capacity with off-plateau
96
relaxation heat capacity. . . . . . . . . . . . . . . . . . . . . . . . . .
6-7 Empty calorimeter AC (no off-plateau correction) and relaxation heat
97
capacity, 0 kG and 1 kG. . . . . . . . . . . . . . . . . . . . . . . . . .
heat
relaxation
correction)
and
off-plateau
AC
(no
6-8 Empty calorimeter
capacity, 3 kG and 5 kG. . . . . . . . . . . . . . . . . . . . . . . . . . 97
6-9 Empty calorimeter low temperature AT(t), showing best fit to sum of
99
two exponentials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
meato
bath
Kb,
conductance
zero
field
thermal
calorimeter
6-10 Empty
sured and predicted from published data on copper. . . . . . . . . . . 99
6-11 Field dependence of Kb. . . . . . . . . . . . . . . . . . . . . . . . . . 100
6-5
6-6
6-12 Sr 3 CuPtO.5 Iro.50 6 calorimeter 136 mK thermal transfer functions. .
6-13 Sr 3 CuPtO. 5 1r o .5 0 6 calorimeter 400 mK thermal transfer functions. .
6-14 Sr 3 CuPtO.5 Iro. 5 0 6 calorimeter AC heat capacity with off-plateau cor-
103
104
rection, and relaxation heat capacity data. . . . . . . . . . . . . . . . 106
6-15 Sr 3 CuPtO. 5 1ro. 5 0 6 calorimeter AC (no off-plateau correction) and re-
laxation heat capacity, 0 kG and 1 kG. . . . . . . . . . . . . . . . . . 106
6-16 Sr 3 CuPtO.5 1rO. 5 0 6 calorimeter AC (no off-plateau correction) and re-
laxation heat capacity, 3 kG and 5 kG. . . . . . . . . . . . . . . . . . 107
6-17 Sr 3 CuPtO.5 1ro. 5 0 6 calorimeter AC heat capacity data above 1 K. . . . 108
6-18 Excess Sr 3 CuPtO.s
5 ro.0 O 6 calorimeter AC heat capacity data above 1 K
(see text). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
6-19 Sr 3 CuPtO.5 I r o . 6 calorimeter zero field thermal conductance to bath
Kb, measured and predicted from published data on copper. . . . . . 110
6-20 Field dependence of Kb, Sr 3CuPtO.5 1ro. 5 0 6 calorimeter. . . . . . . . . 110
111
6-21 High field heat capacity of Sr 3 CuPtO.5 Iro. 5 0 6 and empty calorimeters.
7-1
7-2
7-3
Zero field u/T for Sr 3 CuPtO.5 1r o .5 0 6 , fit below 0.4K. . . . . . . . . . . 114
Zero field u/T for Sr 3 CuPtO. 5 1r o .5 0 6 , fit to all data. . . . . . . . . . . 114
7-5
7-6
6. .
Zero field specific heat for Sr 3 CuPtO.5 Iro.5
0
Low field u/T for Sr 3 CuPtO.5 IrO.5 O6 . . . . . . .
o at 10kG for Sr 3 CuPtO.5 Iro. 5 0 6 , and fit to a =
Specific Heat of Sr 3 CuPtO.5 IrO. 5 0 6 as a function
7-7
Comparison
7-4
of various
paramagnetic
salt
. . . . . . . . . . . . .
115
. . . . . . . . . . . . .
AT3/ 2 + B. . . . . . .
of field at 130 mK. .
115
specific
116
117
heats with
Sr 3 CuPtO.s5 r o.5 O 6 . The dotted line is an extrapolation of the power
law predicted by theory for CPI in the universal regime.
. . . . . . .
123
A-i Thermal circuit for case 1. Temperature Tt is measured with the
bottom calorimeter thermometer, referenced to the bath temperature.
Power Qt0, is applied to top heater or Qbot is applied to bottom heater. 128
A-2 Thermal circuit for case 2. Temperatures is measured with the bottom
(Tt) or top (Tt0p) calorimeter thermometers, referenced to the bath
temperature. Power Q is applied with the bottom heater. . . . . . . . 130
16
C-1 Thermal circuit for two-link Schwall model. All temperatures are referenced to the bath temperature, and T2 (t) is measured. . . . . . . .
136
D-1 Thermal circuit for calculation of Schwall model transfer function. The
heater applies flux q at position 0. The thermometer is located at
position m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
140
17
18
List of Tables
5.1
5.2
6.1
6.2
6.3
K3 Fe(CN) 6 calorimeter K, estimates. K, was determined from a fit of
the transfer function to the two-wire model (2WTF Fit), and a lower
limit on K, was determined by comparison of the data and calculated
two-wire transfer functions with various K. (TF Low. Lim.). . . . . .
Heat capacity data taken with relaxation method. Note that the
empty calorimeter results ("Empty Cal. C") are an upper limit due
to the low LR-400 bandwidth, and are for 68 mg of N grease. The
K3 Fe(CN) 6 calorimeter results ("K 3 Fe(CN) 6 Cal. C") are for 6.44 mg
K3 Fe(CN) 6 and 13.7 mg N grease. The "Fritz C" column is the heat
capacity of 6.44 mg of K3 Fe(CN) 6 from the Fritz paper. See text for
explanation of the "errors" in the Fritz data. . . . . . . . . . . . . . .
82
85
Estimates of empty calorimeter K.. K, was determined from a fit of
the transfer function to the two-wire model (2WTF Fit), and from
the slopes of the power/temperature curves and Equation (6.1) (PT
Slope). Lower limits on K, were determined by comparison of the
data and two-wire transfer functions with various K, (TF Low. Lim.),
and by varying the power/temperature curve slopes appropriately by
one standard deviation (PT Low. Lim.). . . . . . . . . . . . . . . . . 95
Relaxation heat capacity from various thermometer/heater combinations on the empty calorimeter. The notation Cth refers to heat capacity measured with thermometer t, heater h . . . . . . . . . . . . . . . 100
Parameters for Schwall model of empty calorimeter. The values with
an "x" next to them are from fits of the measured AC transfer function to the Schwall model. The Cu Fing. column shows estimated
heat capacity for two sets of copper fingers. The Vespel column shows
estimated heat capacity and conductance for the Vespel pegs. . . . . 102
6.4
Sr 3 CuPtO.5 1ro.5 0 6 calorimeter K,.
6.5
the transfer function to the two-wire model (2WTF Fit), and from
the slopes of the power/temperature curves and Equation (6.1) (PT
Slope). Lower limits on K, were determined by comparison of the
data and two-wire transfer functions with various KS (TF Low. Lim.),
and by varying the power/temperature curve slopes appropriately by
one standard deviation (PT Low. Lim.). . . . . . . . . . . . . . . . . 105
Parameters for Schwall model of Sr 3 CuPtO.5 Iro.5 0 6 calorimeter. . . . . 109
19
K, was determined from a fit of
6.6
Relaxation heat capacity from various thermometer/heater combinations on the Sr 3 CuPtO.5 Iro.50 6 calorimeter. The notation Cth refers to
heat capacity measured with thermometer t, heater h. . . . . . . . . . 109
20
Chapter 1
Introduction
One-dimensional spin systems ("spin chains") have been a subject of interest in condensed matter physics and statistical mechanics for many decades. Originally, such
systems were of interest because theoretical problems in many-body physics, phase
transitions, and critical phenomena were more tractable in one dimension than in
three; hence, it was hoped that a greater understanding of critical phenomena in
our three-dimensional world could be obtained by solving the corresponding onedimensional problem [1],[2]. In the late 1960's and early 1970's., powerful theoretical
methods such as the Renormalization Group were developed and applied to these
many-body one-dimensional problems. It quickly became clear that the hope of "extrapolating" one-dimensional results to three-dimensional systems would not be realized, as the behavior of spin chains exhibited profound qualitative differences from
that of three-dimensional systems. Around the same time, three-dimensional materials whose behavior approximated that of the one-dimensional systems were discovered
in the laboratory, and the study of one-dimensional spin systems became a subject
of interest in its own right. This study continues today, and has been characterized
by a remarkable interplay between theory and experiment, due to the ability of theorists to solve many one-dimensional problems on the one hand, and the ability of
experimenters to produce materials to which these theories apply on the other.
Why do one-dimensional spin systems differ from three-dimensional? Consider
two spin systems, one one-dimensional and the other three-dimensional, in which
nearest-neighbor spins interact with a coupling of strength J. This coupling could
be produced by (for example) direct exchange, superexchange, or dipole interactions.
For temperatures kBT > J, the interaction between the spins is unimportant; in
this temperature range, the systems will behave identically. They will have zero
magnetic heat capacity (in zero field), and their susceptibility X will obey the Curie
Law x = c/T. At thermal energies kBT ~ J the systems will behave quite differently.
The primary factor that distinguishes their behavior is the relative importance of
thermal fluctuations of the spin degree of freedom. Consider a hypothetical Ising
chain (that is, one in which the spin can have only two values, up or down) in which
all spins are aligned (Figure 1-1). If a fluctuation occurs at a single site, the chain is
split into two sections, one in which all spins are up, the other in which all are down.
The energy cost of such a fluctuation is 2J, where J is the coupling between nearest
21
I\ lIT1 II
ll II
Figure 1-1: Ordered Ising chain at T = 0 (top) and Ising chain with fluctuation
(bottom).
neighbor spins. However, since the fluctuation can occur at any of the N sites in the
chain, the entropy gain is k, In N. Hence the free energy change for this fluctuation
is
(1.1)
AF = 2J - kBT inN
For the ordered state to be stable, AF > 0, that is N < e 2 J/kBT. For a macroscopic
sample, this condition is likely not to hold. Hence it is .easy" to introduce thermal
fluctuations into the chain and destroy the order [3].
Figure 1-2: Ordered Ising net at T = 0 (left) and Ising net with fluctuation (right).
The same is not true in higher dimensions. For example consider a two-dimensional net of aligned Ising spins (Figure 1-2) with a fluctuation analogous to the
one-dimensional case. Here, the energy cost is 2Jv/N, and the entropy gain kB In A.
The stability condition is then N < e4J /kBT, which holds for large N. Hence we
see that it is more difficult to destroy the order in higher dimensions.
In fact, due to these fluctuation effects, a one-dimensional system will not exhibit
22
long-range order down to T = 0. On the other hand, short range order does develop
in the chain as it is cooled below kBT - J, and the entropy of the system is gradually
reduced over a wide temperature range through this short-range ordering. Hence the
correlation length (T), that is the length in the chain over which this short-range
order is maintained, is a quantity of great interest in the theory of one-dimensional
spin systems, and increases slowly as temperature is reduced.
Within one dimension, there are other properties of the chain that are important
for determining its behavior. One is the nature of the interaction J between the spins;
in particular, whether it is ferromagnetic (FM) or antiferromagnetic (AF), whether it
depends on the spin directions, and whether it involves only nearest neighbor spins. If
the interaction depends on spin direction, we may have Ising (if J2, Jy = 0, J, # 0),
XY (if J1 = Jy :L 0, JZ = 0), or Heisenberg (if Jx = Jy = J, = 0) spins, all of which
show different behavior. Another important property is the magnitude of the spin.
For example, a spin-J chain behaves quite differently from a spin- chain. In the next
section, I will consider effects of the spin magnitude in particular. This will highlight
the differences between quantum spin chains (with spin near 1) and the more intuitive
classical chains (with S - oc) that are important for an understanding of this work.
1.1
Properties of Spin Chains
Because the theory of Sr 3 CuPtjpIrpO6 assumes a nearest-neighbor Heisenberg de-
scription for the spin interactions, I will also assume that for the chains I describe
below. In the Heisenberg model, the interaction between nearest-neighbor spins is
isotropic. The Hamiltonian is written
L-1
H = JZ
L-1
Si -Sii - pHz E Si
i=O
(1.2)
i=O
where the sum is taken over all sites in a chain of length L (that is, having L sites;
length will always be measured in units of the lattice constant), and the Si are in
general quantum spin operators. For J < 0, we have a Heisenberg ferromagnet; for
J > 0, an antiferromagnet. For simplicity, I will take H, = 0. I will also focus
primarily on the entropy and specific heat. (In this thesis, specific heat is taken to
be n! dT'I, with units J/mol/K, and heat capacity Q/dT, with units of J/K.)
I begin with the classical spin chain. In this case, the spin Si is treated as a classical
vector. While this may seem like an unphysical idealization (real spins are quantum),
the classical approximation already works quite well for spin- at temperatures not
too close to T = 0 [2]. However, it is considered here because it follows intuition
most closely, and so will help to highlight the non-intuitive properties of the quantum
chains. In order to obtain a classical vector from the quantum operator Si, one has
the sense that the limit S -+ oc should be taken, as this will lead to an infinite number
of possible Sz, Sy, S,, as is the case for a classical vector. In order to take this limit,
define the unit operators si = Si/S [4], and note that the commutation relations for
23
these unit operators are, for example,
ss
-
ss =
(1/S)is
(1.3)
Also, set JS2 = Jc; then the Hamiltonian (1.2) becomes
L-1
(1.4)
H = Jc E Si *si+1
i=o
Given the commutation relations (1.3), in the classical limit S -+ oo all spin operators
commute. Hence for the FM case, the ground state (attained only at T = 0) will have
all spins pointing in the same direction, and in the AF case nearest neighbor spins will
point in opposite directions. Excited states can be formed by changing the relative
orientations of nearest neighbors by an infinitesimal amount (in this sense, the T = 0
long-range order for the classical chain is even less stable than that of a quantum FM
chain; i.e., the energy cost of excitation is infinitesimally small, whereas the entropy
gain is still k, ln N). The classical nature of the spin leads to unphysical results for
the entropy and heat capacity; at infinite temperature, the entropy is expected to be
infinite; also, Fisher found for the specific heat [4] (Figure 1-3)
1.0
-Classical
o FM
* AF
0.8
0.6
0.4
-
-
0
-
0.2
00
00
0
0
0
0
CO
K
0
I
0.2
0.4
I
I
0.6
0.8
1.0
kT/J
Figure 1-3: Specific heat of various types of spin chains. The AF and FM chain results
were taken from [5], extrapolated to zero temperature using simple spin wave theory,
and the classical result from [4]. These results are plotted assuming the Furusaki
Hamiltonian (1.2); the Bonner and Fisher Hamiltonian uses 2J rather than J.
24
2
c =1c =- 1 ( 2k,
2 TBsinh
T ~
J
h2
(15
1.3
c(T) does not approach zero for T -+ 0, so the entropy S(T) = f6 c(T')/T'dT'
approaches minus infinity as T -+ 0. Hence near zero temperature, the classical
approximation will not describe any real system and an appropriate treatment will
have to consider the spins to be quantum in some way. Another important observation
about the entropy and specific heat in the classical chain is that they are the same
for both the FM and AF chains.
I turn now to the FM spin-} chain. In this case the Si do not commute, and it is
not immediately clear that the classical ground state, with all spins pointing in the
same direction, will also be the spin-! chain ground state. However, a straightforward
analysis [6] shows that the ground state is indeed the same as in the classical case,
and that the lowest-lying excited states at T = 0 can be described by introducing spin
waves into the chain. A spin wave, or magnon, reduces the total chain magnetization
by one unit, and is a traveling wave with dispersion hw = 4SJ(1-cos ka). A schematic
representation of a FM spin wave is shown in Figure 1-4. For temperatures above
T = 0, spin waves are still expected, although the approximation S_ ~ S on which
the T = 0 analysis relies no longer holds [7]. Also, the absence of order in the id chain
at finite temperature changes the spin wave spectrum. In particular, spin waves with
wave vector k less than the correlation length ((T) will exhibit overdamped behavior,
whereas spin waves with larger k will continue to exhibit oscillatory, traveling wave
behavior [2]. These facts, along with the observation that spin waves do not obey the
superposition principle [6], make it seem unlikely that it will be possible to compute
the specific heat using a Debye-like model with the small wave-vector approximation
w = 2SJk 2 of the dispersion for the T = 0 spin wave. The Debye-like model predicts
c oc v/Y; detailed numerical calculations by Bonner and Fisher [8] confirm that the
specific heat has this temperature dependence at the lowest temperatures, although
the amplitude of the v'T term is a factor of 1.3 smaller than is predicted by spin-wave
theory. The Bonner and Fisher result is plotted in Figure 1-3.
T7TT7TVT
Figure 1-4: Representation of T = 0 FM spin wave. Rather than reducing the
chain magnetization by breaking the chain at a single point, which costs energy 2J,
the magnetization is reduced in a spin wave by tilting all spins slightly off the z
axis, which costs much less energy. The spin wave is a traveling wave in which the
difference in azimuthal angle between adjacent spin sites in the wave is constant.
25
For the AF spin-} chain, the classical ground state is not an eigenstate of H. In
1931, Bethe found the ground state eigenfunction of the spin-! chain, and showed
that it was not ordered: that is. this chain does not exhibit long-range order even
at T = 0 [9]. Given this, the physical nature of the T = 0 excited states is not
clear. Nevertheless, des Cloizeaux and Pearson [10] have computed the dispersion for
the low-lying excited states and found w o( Isin kal, which is the same k dependence
found if one assumes the classical ground state for T = 0 and computes the spin wave
spectrum [11]. However, there are some differences between the true excited states
and the classical ones (for example, the true first excited state is a triplet, whereas
the classical state is a doublet), and the physical nature of the excitations remains
unclear. Again using a naive Debye-like model to obtain the temperature dependence
of specific heat, one expects c cx T (for w c< k dispersion, valid for long wavelengths).
Surprisingly, this temperature dependence is confirmed by the Bonner and Fisher
results, although the amplitude of the linear term is a factor of three smaller than the
spin wave result (experiment also supports the Bonner and Fisher results [1]). The
Bonner and Fisher result is shown in Figure 1-3.
To summarize, the 1D classical model yields the AF and FM ground states that
we intuitively expect, and shows high-temperature behavior that approximates the
behavior of chains with S = 5 and larger. However, its low-temperature behavior is
unphysical, and its thermodynamics is the same for both AF and FM chains. The
spin-- FM ground state is the same as the classical one, and the T = 0 excitations
are spin waves, each of which lowers the total spin of the chain by one quantum. At
temperatures much above T = 0. it is not clear that the simple spin wave picture
is valid; however, it gives a qualitative account of the low-temperature specific heat.
For the spin-j AF, the ground state is not the same as the classical one; moreover,
it is disordered. However, assuming the classical ground state and computing the
resulting spin wave spectrum again leads to an accurate qualitative account of lowtemperature specific heat. For our purposes, the most critical feature of the quantum
picture is the result that the excitations near T = 0 are spin waves, with differing
dispersions for the FM and AF cases.
1.2
Sr 3 CuPt 1 _pIrpO
6
Sr 3 CuPt 1 _pIrpO 6 is one member of a family of new one-dimensional magnetic ma-
terials with general formula Sr 3 MM'O 6 , where M and M' refer to sites in the crystal structure that will accept various magnetic and non-magnetic ions. Work on
Sr 3 MM'0
6
began in 1991 with the discovery of Sr 3 CuPtO6 by Wilkinson et al. [12].
Soon after, Nguyen and zur Loye [13] began a systematic study of the synthesis
and magnetic properties of several members of this family, including Sr 3 CuPtO6 ,
Sr 3 CuIrO6 , and their alloy Sr 3 CuPtlpIrpO6 .
The general crystal structure of Sr 3 MM'0
6
is shown in Figure 1-5. The struc-
ture consists of chains of alternating face-sharing MO 6 trigonal prisms and M'0 6
octahedra, with the chain axis parallel to the c-axis of the crystal. In the ab plane,
one sees a hexagonal net of chains, with each chain surrounded by six clusters of
26
G *
Q
0)
fiU:
06.
k
C9 *
Figure 1-5: Crystal structure of Sr 3 MM'0
6
. Left: view along the length of a chain,
showing alternating MO 6 trigonal prisms (M represented by the small ball at center of
prism) and M'0 6 octahedra (M' represented by the large ball at center of octahedron).
Right: view down the c axis. Clusters of 3 Sr ions surround each chain.
27
three Sr+ ions. Typically, a = 9.6)1, while c = 11.2A (c = 6.7 A for Sr 3 CuPtO6 and
Sr 3 CuIrO6 ) [13] [14]. c is four times the distance between M and M', due to the geometry of the trigonal prism and octahedron. Hence the magnetic ions on the M and
M' sites are at most 2.8A apart, whereas sites on different chains are 5.5A apart. The
separation between these two distance scales leads one to expect that the magnetism
of these materials will be one-dimensional over some temperature range.
Interest in Sr 3 CuPti1 ,IrO 6 in particular was motivated by measurements of Ml/H
at low fields (below 10kG) and at temperatures 2K < T < 300K for Sr 3 CuPtO6 ,
Sr 3 CuIrO6 , and Sr 3 CuPtjpIrpO6 . Nguyen [13] found that NI/H for Sr 3 CuPtO6 fit
well to a one-dimensional AF Heisenberg model, with IJI/kB = 26.1K. For Sr 3 CuIrO6 ,
Nguyen hypothesized that M/H showed one-dimensional FM behavior, with the peak
in MT/H indicating a coupling on the order of J/kB ~ 30K. However, measurements
of M/H on Sr 3 CuPti_pIrpO 6 showed Curie Law behavior down to 2K (Figure 1-6).
This was quite surprising, given that J for the parent materials was around 30K.
The chemistry of Sr 3 CuPtO6 indicates that a Cu2+ ion, with spin-), will be located
in the M site. Cu 2 + generally behaves as a Heisenberg spin- [1]. Pt 4 + is located in
the M' site, and has spin zero. Hence the AF behavior of Sr 3 CuPtO6 can be explained
if two Cu 2+ on nearest M sites interact via superexchange through the Pt 4 + on the
intervening M' site. Since superexchange is always an AF interaction [7], and since
the M-M' distance is only 1.71 in Sr 3 CuPtO6 , this explanation is reasonable.
The chemistry of Sr 3 CuIrO6 leads to a spin-1 Cu 2 + ion occupying the M site in this
material as well. The M' site will be occupied by Ir4+, which also has spin-!. The FM
behavior of Sr 3 CuIrO6 can be explained if the nearest-neighbor Cu 2 + and Ir 4 + ions
interact ferromagnetically via direct exchange, which is reasonable given the Cu2+_
Ir 4 + distance of 1.7A. Magnetization versus field data at 5K for Sr 3 CuIrO 6 shows
that only one-third of the full moment 2 p, is obtained for fields up to 20T in powder
samples [15].
If Sr 3 CuIrO6
were a Heisenberg FM, there would be no preferred
direction and it would be possible to achieve the full moment in the powder samples
at reasonably low field. This result suggests that there is some strong anisotropy in
Sr 3 CuIrO6 that restricts the spins to align along a particular direction in the crystal.
One possible reason for this anistropy is ferrimagnetic ordering of Ising-like spins
on a hexagonal net. Such ordering could occur in Sr 3 CuIrO6 if a strong, AF, Ising
coupling existed between short FM segments in different chains. Magnetization data
on Ca 3 Co 2 0 6 [16], which is isostructural with Sr 3 CuIrO6 , has been interpreted in
terms of this type of Ising ferrimagnetism. Of course, if ferrimagnetism is the correct
explanation of the M versus H data, then Sr 3 CuIrO6 is not one-dimensional below
5K.
On the other hand, this does not mean that Sr 3 CuPt0 .5 Ir 0 .5 0 6 could not be
one-dimensional below 5K. In any case, possible consequences of such anisotropy for
Sr 3 CuPt0 .5 Ir 0 .5 O6 will be discussed further in Chapters 2 and 7.
Given the magnetic structure of Sr 3 CuPtO6 and Sr 3 CuIrO 6 , we expect the alloy
CuPtj.pIrpO
Sr 3
6 to consist of spin-! chains with a random distribution of nearest-
neighbor FM and AF bonds, with probability p that a given bond is FM, 1 - p that it
is AF. This model has been considered theoretically by P.A. Lee and his collaborators
in a series of papers, and will be discussed in detail in Chapter 2. This theory was
found to give a good account of the unusual Curie Law behavior (Figures 1-7 and
28
Sr 3 CuIrO6
Sr 3 CuPtO6
0.006.
0.6 -
I =:.Z1:u6.1 K
0.005 -
K
S
0.004 -
0 0.4
-
0.003 S
0.2
0.002 -
S
I-D Halse::er3 Modei
0.001
6
50
100
150
:00
Temperaturz
250
Sp*------
0.0
300
0
5b
100
K
150
200
250
300
Temeraure(Kj
Sr3 Cu Pt 0 .I r05 0 6
0.04
0-
E 003
0
E 0.02
*
0.01
-
*@*00...
0
0
Figure 1-6:
10 20 30 40
Temperature [K]
50
M/H data for (top left) Sr 3 CuPtO6 , (top right) Sr 3 CuIrO6 , and
Sr 3 CuPtO.5 IrO. 5 0 6 (bottom).
29
1-8), and made several predictions about the low-temperature (< 2K) properties of
the alloy. In particular, the alloy was predicted to have an unusually large spin
entropy content at low temperatures (10 - 30% of R ln 2, depending on p), and at
the lowest temperatures to exhibit behavior characteristic of a new universality class
of random quantum spin chains, with characteristic scaling laws in specific heat and
susceptibility.
800
*
Sr 3 CuPtO6
A
Sr 3CuPto.75Iro.2506
I
Tv Sr 3CuPto.soIro.5006
600
+
Sr3CuPto.:Iro.75Ose6
*
SrsCuIrO6
*
A
A
F
A
*V
A
400
A
A
V
I
A
V
U
A
A
A
V
200
A*
VT
:
T
.MMMMU
0
0
50
100
150
200
250
Temperature (K)
Figure 1-7: 1/x data for Sr 3 CuPtl-,IrO 6 taken by Nguyen [15].
30
300
20.0
15.0
h
10.0 -
5.0
p=1
0.0
0.0
4.0
8.0
4k BT/J
Figure 1-8: 1/x theoretical prediction of Lee and collaborators [17].
31
1.3
Specific Heat of Sr 3 CuPt0 5. 1r0 .o5 0 6 below 1K
The goal of this work is to investigate the specific heat of one particular alloy,
Sr 3 CuPt 0 .5 Ir 0 .5 0 6 , at temperatures below 1K.
Of course, one reason to measure
specific heat of Sr 3 CuPto.5 Iro. 5 0 6 below 1K is to discover whether it is a material that obeys the theory. However, there is another reason for interest in
Sr 3 CuPto.5 Iro. 5 0 6 (and Sr 3 CuPtipIrpO6 in general).
If it does obey the theory, or
at least is found to have a substantial low-temperature entropy content, the material
may be useful for refrigeration via adiabatic demagnetization. Adiabatic demagnetization of paramagnetic salts was the only technique available for refrigeration below
300mK until the early 1970's, when dilution refrigerators became widely available.
However, most dilution refrigerators do not provide access to temperatures below
10mK, and for temperatures below 1mK the only known refrigeration technique is
nuclear demagnetization [18]. The hope is that Sr 3 CuPt 0 .3Ir0 .3O6 may prove superior in some way to dilution refrigerators or paramagnetic salts in the 10mK - 1K
range, or that it may even provide competition for nuclear demagnetization in the
low millikelvin to 100pK range.
Sr 3 CuPt0 5 Ir 0. 5 0 6 is focused on for two reasons: first, the most extensive experi-
mental work thus far has been done on Sr 3 CuPto.5 Iro. 5 0 6 ; second, the distribution of
FM and AF bonds in this particular alloy is (in principle) completely random. Specific
heat is measured for several reasons. First, no other groups have yet reported specific heat measurements on Sr 3 CuPt 0 .5 Ir 0 .5 0 6 below 1K. Second, specific heat gives
the most direct access to entropy content. Third, if Sr 3 CuPt 0 .5 Ir 0 .5 0 6 undergoes a
transition to long-range (two or three-dimensional) ordering at some temperature,
specific heat provides a definitive signature for such a transition. Finally, if there are
phases in addition to pure Sr 3CuPto.5 Iro. 5 0 6 or magnetic impurities of some kind in
the measured Sr 3 CuPto.5 Iro.05 6 sample, these are likely to have lower entropy content
than Sr 3 CuPt 0 .5 Ir 0 .30 6 . Therefore the presence of such phases should not confuse the
comparison of the specific heat data with theory.
In order to achieve this goal of specific heat measurements on Sr 3 CuPt0 .5 Iro. 5 0,
it was necessary to design, build, and test a new experiment for specific heat measurements below 1K. The resulting apparatus allows for specific heat measurement
from 100mK to 2K in magnetic fields to 7T using either AC or relaxation calorimetry.
While based on earlier designs, the apparatus has a few unusual features. In particular, it allows for accurate AC calorimetry below 1K, incorporates a non-destructive
sample mounting technique, and uses a new arrangement for relaxation calorimetry. The experiment was tested by measurements of the field-dependent specific heat
of potassium ferricyanide K3 Fe(CN) 6 , a common material used in blueprinting and
photography that behaves as a one-dimensional Ising AF below 1K and has a (threedimensional) Nel transition at 130mK.
This work is organized as follows. In Chapter 2, I discuss the theory of random
quantum spin chains, developed by P.A. Lee and his collaborators, and experimental
work to date on Sr 3 CuPti-IrpO6 . In Chapter 3, I describe in detail the two methods used to measure specific heat, and mathematical models for these methods. In
Chapter 4, the design and construction of the apparatus is described. In Chapter
32
5, measurements and results on K3 Fe(CN) 6 are described, and in Chapter 6, measurements and results on Sr 3 CuPtO.*
5 ro. 5 0 6 . Finally, in Chapter 7 I conclude with a
presentation of the final results for specific heat of Sr 3 CuPtO.5 1ro. 5 0 6 , discussion of
the results, and recommendations for future work.
33
34
Chapter 2
Random Quantum Spin Chains
and Sr 3 CuPtipIrpO6
2.1
Theoretical Work on RQSC
The statistical mechanics of random quantum spin chains has been studied by
Lee and collaborators [19] [5] [20] [21] [22] in a series of papers. They modeled
Sr 3 CuPtIIrpO6 as a spin-! Heisenberg chain with Hamiltonian
L--
'=
L-1
JiSi - Si+ - MHz E SZ
i=O
(2.1)
i=O
The Ji are given by the probability distribution
P(Ji) = p3(Ji + J) + (1 - P) (Ji - J)
(2.2)
This distribution indicates that a given nearest-neighbor bond will be of strength J
and have probability p of being ferromagnetic and probability 1 -p of being antiferromagnetic. Hence the chain can be pictured as a collection of connected ferromagnetic
and antiferromagnetic chain segments, where the typical lengths of the FM and AF
segments depend on p.
Furusaki et al. applied high-temperature expansion [23] [24] and transfer matrix
methods [25] [26] [27] to analyze this system, and postulated the following physical
picture based on their numerical results. For kT > J, the spins are decoupled and
the system behaves as an ordinary paramagnet. For kBT < J, FM or AF correlations grow within the segments. Excitations of very short segments (a couple spins
in length) will have energies of order J, and so will not contribute significantly to the
thermodynamics in this temperature range. Longer segments can be described in a
spin wave picture, with a spin wave population in each segment determined by kBT,
the length, and the type (FM or AF) of segment. Since FM and AF segments have
different spin waves dispersions, a spin wave from a FM segment will not propagate
into the adjacent AF segments (and vice vesa) (Figure 2-1). Hence the thermodynamics of the system in this temperature range is the same as that of a collection of
relatively short, decoupled segments. Since the segments are relatively short, there
35
is a substantial energy gap between the ground (zero spin wave) state and the first
excited (one spin wave) state. Hence for kBT < J, these decoupled segments will
be found in their local ground (zero spin wave) state, which is the state of maximal
(minimal) total spin for a FM (AF) segment. In this low temperature regime, a FM
segment containing n sites (for example) therefore behaves like a single "large spin"
of spin n/2.
Figure 2-1: Schematic view of spin wave confinement. Due to different dispersion
relations for FM and AF segments, spin waves from a given segment do not propagate
into adjacent segments. A mechanical analogy is a rope with discontinous changes in
thickness.
One might wonder whether the presence of these decoupled. large spins could be
detected via a change in. for example, the Curie constant c = xT. The argument is
most easily made for classical spins of magnitude So at each site rather than quantum
spins-! as is the real case for Sr 3 CuPt 1 -IrpO
6
; however, the argument holds for the
quantum case as well [17]. For kBT > J, the susceptibility per site x/N is just
2S2
x/N
=
2
3kBT
(2.3)
3kBT
For kBT < J, short range order forms, with neighboring spins locked either parallel or
antiparallel to each other (depending on whether their bond is FM or AF) in clusters
of length (the correlation length). Note that these clusters will contain both FM
and AF bonds-in the classical case, the spin wave picture does not apply and so there
is no decoupling of FM and AF segments. In fact, a simple change of variables maps
the classical problem to a simple classical FM chain without changing the energy
spectrum [19]. However, these clusters form an effective spin Seff. To compute Seff
for p = 0.5, one moves from one site to the next in the cluster, adding the spin at
each site to Seff. For p = 0.5, there is an equal probability of increasing or decreasing
Seff by one unit at each site. Therefore (S'2f) can be computed in a random walk
36
picture, with the result (Sjff) = So. In this case the total susceptibility is
x
-
N /p (2n f)
B(Sf
3kBT
(2.5)
Hence the susceptibility per site is
xN
(BSo)
2
(IB So) 2
3kBT
(2.6)
(2.7)
The Curie constant is the same for the two temperature regimes. For a quantum spin
S, the Curie constants are slightly different- for kT > J, S2 is replaced by S(S +1)
above, and for kBT < J So is replaced by S [17].
Furusaki et al. found that their numerical methods began to fail at an energy scale
kBT
-
J/5, and that the entropy per site obtained by integrating C/T from infinite
temperature down to J/5 was less than the total entropy per site kBln2. The amount
of this "missing" entropy varied with p: 12%, 24%, and 36% for p = 0.25, 0.5., 0.75
respectively. As a test of their physical picture of decoupled segments, they assumed
that this picture applied at J/5 and computed the amount of entropy in the decoupled
segments. A match between this amount and the amount "missing" from the HTE
and TM calculations would lend credence to the decoupled segments picture.
In order to compute the amount of entropy in the decoupled segments picture,
Furusaki noted that a FM segment containing I bonds has entropy kB In 1, an AF
segment with I odd has zero entropy, and an AF segment with 1 even has entropy
k, ln 2. (Furusaki determined that the boundary spins, which are shared by adjacent
FM and AF segments, must be assigned to the AF segment.) Then the entropy
contribution per site is given by
S= E
N
(2.8)
where the sum is over all segments, and N is the total number of sites in the chain.
This can be rewritten
S
+
ZAFsegs nAF +
AFsegs SAF
ZFMsegs SFM
FMsegs nFM
(2.9)
that is, the sums are taken over the AF and FM segments separately. Noting for
example that EFMsegs nFM
=
(nAF)N,AF, where Ns,AF is the number of AF segments,
and that the number of FM segments is equal to the number of AF segments (they
may differ by 1, which can be neglected for a chain with a large number of sites),
+ (SM)(2.10)
(rAF)+ (rFM)
S(SAF)
37
Given the probability densities for FM and AF segments with 1 bonds,
PF(1)
=
PA (l)
=
(2.11)
(2.12)
- P)P'1
p(1 - p)'
the quantities in (2.10) are found to be
00
(SAF)
(2.13)
A (2m)
ln(2S +1)
m=1
(2.14)
PF(l) ln[2S(l - 1) + 1]
(SFM)
11
1
(nAF)
+p
(2.15)
p
(nFM)
(2.16)
i-p
From these, s can be computed as a function of p. Good agreement is found between
s(p) found in this way and the amount of missing entropy found by Furusaki et al.
from their HTE calculations (Figure 2-2). Hence the decoupled segment picture seems
to provide a reasonable model for this low-temperature (kBT < J/5) regime.
I
I
I
I
I
I
'
0.4
0
Model
o HTE
U.3
0.2
<1
0.1
0
I
0
I
0.2
I
I
I
0.4
0.6
0.8
I
1.0
P
Figure 2-2: Missing entropy as a function of p, calculated with (line) Equation 2.10
and (open circles) HTE.
Since there is no frustration in this disordered system, one expects all of the
missing entropy to be removed from the system between kBT
38
-
J/5 and T = 0 .
Furusaki et al. proposed that the coupling between segments, while much smaller
than J, is not zero. To test this proposal, they carried out exact diagonalization of
Hamiltonians of finite length chains (around 10 spins) with various configurations of
FM and AF bonds, and found that in the ground state adjacent segments were indeed
coupled. Accordingly, the lowest-lying excited states were intersegment spin waves,
that is spin waves in which total Sz, summed over all large spins in the chain, was
reduced by 1 for each spin wave. These low-energy spin wave excitations were of course
at much lower energy than spin wave excitations within an individual segment, where
S, of an individual large spin is reduced. Hence the "missing" entropy remaining
below J/5 would be removed from the system by correlations between the segments
that arise at much lower energies (temperatures) than the correlations within the
segments. The sign and strength of the couplings between segments depended on the
ordering (FM or AF) and size of the adjacent segments.
This work on exact diagonalization of finite chains motivated Westerberg et al. to
consider the Hamiltonian
'=
JiSi
(2.17)
where the magnitude and sign of the Ji and the size of the spins Si are random
(Figure 2-3). Here, the Si represent the large spins discussed above. Westerberg
et al. took this Hamiltonian to model the low-energy properties of the RQSC, and
attacked it using a generalization of the real-space renormalization group (RSRG)
method developed by Dasgupta and Ma [28] for the study of random AF spin-!
Heisenberg chains. The physical rationale for the RSRG is that at a given temperature
T, the spins coupled by J, > kBT will have frozen into their local ground state, and
the thermodynamics will be determined by spins coupled by Ji < kBT. Hence as the
chain cools, the spins remaining to participate in the thermodynamics will be coupled
by progressively weaker Ji, and the distribution of spin sizes and bond strengths will
change (see Figure 2-4).
1
2
1/2
5/2
0
6
0
Figure 2-3: Spin chain considered by Westerberg et al. (Equation 2.17). The FM
segments form large effective spins, separated by AF segments of variable length.
39
SL
S1
A1
SR
A0
S2
A2
RENORMALIZE
SI
SL+ SR
A1
S2
A2
Figure 2-4: Schematic of the RSRG method. A renormalization step proceeds by first
identifying \O, the strongest bond in the chain. Then the two spins linked by .\o are
frozen to form an effective spin SL +SR, and the nearest neighbor bonds renormalized
to i\1 and Z12 . As the chain cools below kBT - Ao, it effectively carries out a step in
the RSRG.
40
They found that the distributions of spin sizes and bond strengths in the chain
flow to fixed-point distributions as the RSRG proceeded. From these fixed-point
distributions of spins and bonds, scaling laws governing the thermodynamic properties
of the system in the universal regime can be computed. For finite temperature T,
large spins coupled by bonds weaker than kBT can be considered to be free spins,
since in the universal regime the bonds are typically much weaker than kBT. Hence
the (zero-field) entropy per site will be given by
s(T, H = 0)/L oc keln(2(S) + 1)/n
(2.18)
where (S) is the average spin size (computed from the distribution found via RSRG),
and n the number of spins-! per large spin. In the universal regime, they found
n ~0 2, where A0 is the strongest bond in the chain and a = 0.22 ± 0.01. Also,
(S) ~ A . Hence
s(T, H = 0)/L oc akBT 2, | InT
(2.19)
and for the specific heat
-(T)/L oc| InT IT
(2.20)
The effect of an applied field H will depend on how the magnetic energy p(S)H
compares with kBT. If p(S)H < kBT, the field will not be able to overcome the
thermal fluctuations of the free spins, and the specific heat will still be given by
(2.20). On the other hand if p(S)H > kBT, then the field will start to align spins
with couplings of A0 ~ p(S)H or less. Using the scaling result for (S), the energy
scale at which the field starts to align spins is thus AH ~ H 1/(1+"). Hence if one were
to study, as a function of applied field, the temperature at which the entropy started
to decline rapidly, one would find the relation T ~ H1/('+").
The work of Westerberg et al. provides at least two methods to search for evidence
of universal behavior via specific heat. First, one could simply measure specific heat
as a function of temperature in zero field and look for the expected dependence given
by equation (2.20). Second, in an applied field one should find a peak-like structure
in the specific heat at a temperature corresponding to the energy scale Ao' described
above. By studying the way in which the temperature at which this peak occurred
varied with field, one could attempt to verify the relation T ~ H1/(1+c).
Further theoretical work on the RQSC was done by Frischmuth and Sigrist [22].
They applied the continuous time quantum Monte Carlo loop algorithm [29] to the
Hamiltonian (2.17) and assumed an initial distribution
P (J)=
1 -J 0 < J < Jo
0, otherwise
2J'
This initial distribution was chosen so that the algorithm would be accurate well
into the universal regime and because Westerberg's theory should apply to it. Hence
Frischmuth and Sigrist's results provide an independent check on Westerberg et al.'s
work. Frischmuth and Sigrist found that their results agreed very well with those
of Westerberg; in particular, they found a = 0.21 ± 0.02, whereas Westerberg found
41
a = 0.22 ± 0.01.
The theory presented above provides a consistent description of systems with
Hamiltonian given by (2.1). Experimentally, it is critical to know whether and how
the theory applies to Sr 3 CuPtiJIrpO6 in particular. Note that the Hamiltonian differs from Sr 3 CuPt1_JIrpO 6 in at least four ways: first, the magnitude of the Ji are
different for FM (50K) versus AF (26K) bonds; second, the FM bonds always occur
in pairs; third, in light of the high-field magnetization data on Sr 3 CuIrO 6 (Chapter 1),
it is not clear that it is appropriate to treat the FM segments as Heisenberg; finally,
couplings between the chains are neglected. Regarding the first two differences, it
is thought [5] that bond randomness is the factor that will determine the physics of
Sr 3 CuPtjJIrpO6 . Hence, within certain limits, the quantitative details of the bond
distributions will not change the basic physical description of the system as given by
the model. Regarding the third difference, the theory requires that some kind of spin
wave picture can be used to describe the FM segments. This will be the case if the
FM interaction is Heisenberg or XY [30], but not if it is Ising. The fact that the theory gives an account of the high-temperature data (see Figures 1-7 and 1-8) suggests
that a spin wave picture does apply, at least in that temperature range. More measurements may help to clarify the situation. For example, high-field measurements
on Sr 3 CuIrO6 well above 30K would be desirable as a control on the 5K result, as
would high-field measurements at high and low temperature for Sr 3 CuPtO.5 1r o.5 0 6 it-
self. As for the fourth difference, since the distance between chains is large (10A),
wave functions of electrons on different chains will not overlap directly. Hence the
chains will not interact via exchange interactions. A band structure calculation (using
the extended Hiickel method) for an isolated NiPtO6- chain and for Sr 3 NiPtO6 [31]
shows that the band structure for an isolated chain is virtually identical to a chain in
Sr 3 NiPtO6 . It also shows that there are very few Sr states available at the energies
occupied by the unpaired Ni electrons. The Sr is therefore unlikely to perturb the Ni
electron wave function, so no superexchange coupling via the Sr is expected. Hence
the strongest interchain interaction will be the magnetic dipole interaction. This can
be estimated via [6]
a
1
U ~
(2.21)
13 7 ()(Ry
where r is the distance between moments and Ry the Rydberg constant. Naively
plugging in the interchain distance for r, U ~ 1OmK. Hence it is at least plausible
that Sr 3 CuPtJIrpO6 will remain one-dimensional down to the lowest temperatures
accessible by a dilution refrigerator.
Pursuing the question of couplings between chains further, the temperature T,
at which a system consisting of one-dimensional chains is expected to undergo a
transition to long-range, three-dimensional order is given by [2]
kBT ~ J's2 d(T)
(2.22)
where s is the spin at a site in the chain, J' the coupling between chains, and 1d
the correlation length. This result is obtained from the simple argument that as
correlations grow within the chain, the correlated sections can be treated as a single
42
large spin of magnitude s 1d, and that three-dimensional order will occur when the
interaction between two such large spins in adjacent chains is comparable to the
thermal energy. This "amplification" of T, by 1d should be quite large for p = 0, 1,
whereas for intermediate values of p the bond disorder will limit 1d. Hence one might
expect an interesting dependence of T, on p, which might give one some indication of
the sizes of the large spins in the chains at the given p.
Ramirez has measured low-temperature specific heat on Sr 3 CoPtO6 [15], in which
cobalt has spin 2 and platinum spin zero or spin 1. This material was thought to be
a possible example of a classical random spin chain, for which Lee and collaborators
have also made predictions [19]. Ramirez found a long-range ordering transition at
1.4K [32]. Using the classical expression for 1d [2], and the measured Tc, I find
J _ 4mK. This result lends further support to the conjecture that the coupling
between the chains is very weak. I emphasize that for a classical chain, as discussed
above, the disorder does not limit the size of (T), so one would expect a higher T,
for the classical system than for Sr 3 CuPt 1 _pIrpO 6 .
However, even if there are no materials properties of Sr 3 CuPt 1 _,IrpO 6 that obvi-
ate its description by the theory, the energy scale at which the actual distributions
of bonds and spin sizes for Sr 3 CuPt 1 _pIrpO 6 begin to match those of the universal
distribution may be experimentally inaccesible. Westerberg et al. attempted to address this issue by beginning the RSRG procedure with a distribution simulating
Sr 3 CuPt1pIrpO6 for p = 0.8. They found that the approach to the universal distribution was very slow; indeed, (S) and (n) do not show pure scaling behavior until
Ao/J ~ 101. For J ~ 40K, this leads to temperatures clearly below anything
available experimentally for condensed matter systems. Nevertheless, measurements
in the experimentally accessible temperature range are of value as a test of the theory
(for example, what if scaling behavior was observed at an accessible temperature?),
and to determine the true low-energy physics of Sr 3 CuPt 1 _,IrpO 6 (for example, it may
undergo a phase transition to long-range, three-dimensional order).
2.2
Experimental Work on Sr 3 CuPt1_pIrpO 6
As mentioned in Chapter 1, DC magnetic susceptibility for temperatures 2K < T <
300K has been measured for Sr 3 CuPt1_pIrpO 6 with various p and compared with the
theory of Furusaki et al.. Excellent qualitative agreement between the kBT > J/5
theory of Furusaki et al. and experiment was found [17]. (see Figures 1-7 and 1-8).
These results encouraged further experimental study of Sr 3 CuPt 1iIrpO
6
, particularly
at lower temperatures where the decoupled segment picture and the theory of Westerberg et al. might apply. AC susceptibility of p = 0.5, 0.667 samples has been measured
on 50mK < T < 30K and in fields OkG < H < 2kG by Beauchamp and Rosenbaum.
Also, specific heat of a p = 0.5 sample has been measured on 2.5K < T < 50K in
zero field by Ramirez [15]. I discuss these measurements below.
Beauchamp and Rosenbaum (BR) found a broad peak in x'(w) at 1.7 K for frequencies of 200Hz, 500Hz, and 3000Hz in both p = 0.5 and p = 0.667 samples (see
Figure 2-6). The peak is suppressed with increasing frequency and field. BR also
43
measured magnetization as a function of field M(H) for both p = 0.5 and p = 0.667,
and found results shown in Figure 2-5. These results indicate that M(H) begins to
saturate at fields as low as 100 gauss at 5K. Assuming that M(H) can be described
by a Brillouin function, and noting that for the Brillouin function saturation begins
for pagH(2j+1)/2kBT ~ 1, 1 find j ~ 400. This result is inconsistent with Furusaki's
theory, for which one would expect j of order 1.
Further AC susceptibility measurements to 0.3 K were done very recently on
new samples of p = 0.3, 0.5, 0.7 material by Beauchamp. For all three, a peak was
observed in x' around 1.7 K, with the peak temperature increasing slightly with
p. The peak is much broader than would be expected for a long-range ordering
transition [34]. Moreover, the peak is strongly suppressed by low (0.5kG) fields, and
the peak temperature does not appear to shift very much if at all (Figure 2-7). If this
peak represented long-range ordering with a coupling J ~ 2K, one would expect to
need fields of 10kG or more (kBT ~~J ~ pH) to alter the peak substantially, and
would expect the peak to shift in temperature with increasing field. Overall, these
measurements suggest some qualitative change in the behavior of the material, but
this change is unlikely to be a transition to a three-dimensional, ordered ground state.
Sigrist [33] has proposed a model to explain the results of Beauchamp and Rosenbaum. According to this model, there are (non-statistically) long FM sections (referred to by Sigrist as "FM clusters") in the chain, which may have arisen from
inadequate mixing during sample fabrication. These clusters correlate to form very
large effective spins at temperatures below the exchange energy of 40K. The presence
of these large effective spins is the cause of the large j observed in M(H) at 5K.
Around 1.7K, large effective spins on the same and different chains begin to correlate due to competing AF and FM interactions between them. A spin system with
competing interactions (frustration) and some kind of disorder can form a spin-glass
state [61]. Hence Sigrist attributed the broad peak at 1.7 K to a spin-glass transition
at which these FM clusters correlate.
The Sr 3 CuPtl._IrpO6 samples (p = 0.5 and p = 0.8) used for this study were subjected to repeated grinding and firing cycles in hopes improving the randomness of the
bond distribution. However, these samples showed the same M(H) behavior found by
BR (Figure 2-8). Ramirez and collaborators [15] found no indications of long-range
ordering in the specific heat of Sr 3 CuPtO.5 1r o .5 O6 down to 2.5K (Figure 2-9). Comparisons between Sr 3 CuPtO.5 1r o .5 0 6 and diamagnetic Sr 3 ZnPtO6 indicate additional
weight in the spectrum below 20K, possibly due to magnetism in Sr 3 CuPtO.5 1r o.5 0 6 .
However, it is clear that access to lower temperatures would be necessary to make
meaningful comparisons with the theory. The goal of this thesis is to measure specific
heat of Sr 3 CuPtO. 5 Iro.50
6
at these lower temperatures.
44
40
I
I
300
7
.20
--- x=0.667
- -= -x=0.5
C-
20'20
-1o0 1
10
-20
0
7
-U
10 0
K
-U-
'
I
0
-500
T=5 K
0
500
1000
1500
2000
I
2500
H(G)
Figure 2-5: 1I versus H data of Beauchamp and Rosenbaum on p = 0.5, 0.667
samples.
45
0.40*
-
0.30
E
0.20
*o
x=0.667
-.
x=0.5
0.10
*
L
-)
U
0.00C
)
5
10
15
T(K)
20
25
Figure 2-6: AC susceptibility data of Beauchamp and Rosenbaum.
46
30
0.08
Magnetic Field
-.
-o--
0.06
6
0.04
OT
0.025 T
0.05 T
-- +0.5 T
0.2 T
-~--05T
01 T
u
~'
000
0.02
0
5
15
10
20
25
30
T (K)
Figure 2-7: Recent AC susceptibility data on Sr 3 CuPtO.5 Iro. 5 0 6 taken by Beauchamp.
47
60
50
-
A
p=Q. S 20K
7
p=O.S 5 K7
7
-
-
40
CU
-
30
E
-7
E
a)
20
-
-
'V
10
0
10
0
200
400
600
Magnetic Field [Gauss]
800
1000
Figure 2-8: Magnetization versus field for Sr 3 CuPto.5 Iro.0 O 6 samples used in this
thesis.
48
T
0.8
V
A
'9
SraZCPtO,
sracopto,
a a
V enpy
0.6
A*
4
SeCu(FPtyO,
-
le1
0.4 --
0.2
9 V V
7
7
777VV7
7
0.09
0
0.050
10
30
20
40
50
T (K)
Figure 2-9: Heat capacity data of Ramirez. In each case, the sample consisted of 0.2g
of the Sr 3 MM'0 6 material, and 0.2g of silver powder, which were compressed together
to form a pellet. The pellet was then glued to the calorimeter [32].
49
50
Chapter 3
Methods
3.1
Quasi-Adiabatic Calorimetry
Heat capacity at constant field is defined as
C = (4Q/dT)H
(3.1)
Operationally, C can be most simply measured by applying a known heat to a sample
and measuring the resulting temperature change. This is known as adiabatic calorimetry, so-called because the sample must be thermally isolated from its surroundings
in order to obtain an accurate measure of 4Q. In order to use this technique at low
temperatures, some kind of thermal switch must be used to connect the sample to a
source of refrigeration. The switch is closed while the sample is cooled to the desired
temperature, then opened for adiabatic calorimetry measurements. Several types of
switches have been used successfully, including gas, mechanical, and superconducting
types [18].
When trying to measure the specific heat of a new material, one generally has only
a small (less than 1g) quantity of material available. Adiabatic calorimetry becomes
very difficult for such small samples at low temperatures. Since low-temperature
specific heats are small, small samples have low heat capacity, and so are very sensitive
to heat leaks (dTeak = 9Qleak/C). These heat leaks could be due to conduction down
electrical wires connected to thermometers or heaters on the sample, to radiation
from warmer surfaces near the sample, and/or to residual exchange gas. As a result,
the sample may warm to unacceptably high temperatures, and could undergo large
temperature fluctuations if the heat leaks vary in time. The sample can be cooled and
its temperature stabilized against the effects of heat leaks by increasing its thermal
contact with the refrigerator, but of course this makes an accurate determination of
4Q impossible.
Various solutions have been proposed for this tradeoff between thermal isolation
and the need to cool and temperature control the sample. The main strategy is
to deliberately make a weak thermal connection between sample and refrigerator.
This thermal connection is chosen to be strong enough to provide sufficient sample
cooling and temperature control to overcome heat leaks, yet weak enough so that the
51
thermal conductance across the sample is still much greater than that between sample
and refrigerator. Under these conditions, various quasi-adiabatic techniques may be
applied to measure sample heat capacity. Below I describe the two quasi-adiabatic
methods, thermal relaxation and AC, which were used to measure specific heat in
this thesis. I also discuss mathematical models for these two methods. These will be
used later to describe the behavior of the calorimeters used in this thesis.
3.2
Thermal Relaxation Calorimetry
Thermal relaxation calorimetry was first proposed by Bachmann et al. [35] in 1972,
and has subsequently been adopted by several groups for low-temperature calorimetry
of small samples. [36] [37] A constant power P is applied to the sample until a steadystate condition is achieved. Due to the weak link between sample and refrigerator
(the latter kept at fixed temperature), a temperature difference AT(Po) arises between
sample and refrigerator. If P is turned off suddenly, AT decays exponentially to zero,
with the time constant -r1 for the decay given by
r1 = C/Kb
(3.2)
where Kb is the thermal conductance of the weak link. TF can be determined by
recording AT(t) and fitting to an exponential. If one applies various powers P and
records the steady-state temperature difference AT(P), one can obtain Kb as the
slope of the power/temperature curve P versus AT(P). Given T and Kb, C can then
be determined.
The above method works well provided the minimum conductance in the calorimeter Kmin is much greater than Kb. Ideally, one designs a calorimeter in which all of
parts of the calorimeter are in good thermal contact with each other and with the
sample; in that case, Kmin could be the sample conductance K8 , if the sample were
- Kb, parts
a poor thermal conductor (as Sr 3 CuPt1_pIrpO 6 is). However, for Kmin ~
of the sample that are further from the link will see a significantly smaller thermal
conductance to the refrigerator than those closer to the link, so that AT(t) can no
longer be characterized by a single time constant T1 . In general, one would expect
that under these circumstances AT(t) would be given by a sum over (infinitely many)
exponentials. However, by solving the heat equation for their geometry, Bachmann
et al. found that under certain conditions, only the first two terms in such a sum are
needed to describe AT(t). The time constants of these two terms were r = C/Kb as
before, and
T2
oc C/Kmin. Hence AT(t) which were best described with two exponen-
tials were said to exhibit "r 2 effect". The required conditions for the two-exponential
description are that the thermal link and addenda heat capacities be only a few percent of the sample heat capacity, and that Kmin > Kb/2. In this case, it was possible
to extract the heat capacity (and Kmin) given T(t) and Kb.
Pursuing Bachmann's result further, Schwall et al. noted that a simple model
(Figure 3-1) consisting of two discrete heat capacities with thermal links between
them and to the bath could be solved exactly, and that the solution would be a sum
52
AT 1
C1
Kmin
C2
AT 2
Kb
Figure 3-1: Thermal circuit for Schwall model.
53
T
Kb
I
tot
->'
I
Figure 3-2: Thermal circuit for Sullivan and Siedel model. The calorimeter and
sample are assumed to be in excellent thermal contact, so that the slab with heat
capacity Ctot represents the entire sample/calorimeter assembly. Heat flux j = (e")
(a is slab cross-sectional area) is applied at one end of the calorimeter by the heater,
and temperature T = Tdc + Tac is sensed at the other side of the calorimeter. Heat is
dumped to thermal ground (the heat bath) through the weak link conductance Kb.
of two exponentials, such as was seen in Bachmann's model when the calorimeter
thermal properties met the conditions mentioned above. In the Schwall model, the
decay AT(t) = AT 2 (t); that is, the thermometer is on lump 2. The heater location
is irrelevant, since both lumps are at the same temperature at t = 0 when the heater
is turned off. One then measures AT 2 (t) and fits it to the expression
AT 2 (t) = A 2 exp(-t/TA) + B 2 exp(-t/TB)
(3.3)
to find A 2 , TA, B 2 , and TB. Kb is again the slope of the power/temperature curve.
Then solving the model yields
Kb
CAot = C1B+TC2
(3.4)
That is, the total heat capacity Crt of the calorimeter (sample + addenda) is just
Kb times the weighted average of the two time constants. Schwall et al. found that
they could reliably extract the sample heat capacity by application of their model
solution to determine the total heat capacity Ctot, and then subtracting from Cot the
(separately measured) heat capacity of the addenda. Schwall's approach will be used
in this thesis to extract heat capacities in situations where AT(t) is best described
by a sum of two exponentials.
3.3
AC Calorimetry
Sullivan and Seidel first proposed AC calorimetry in 1968 [38]. The specific geometry
they considered is shown and described in Figure 3-2.
For their analysis, they
assumed excellent thermal contact between all addenda and the sample, and that
the dominant thermal resistance and heat capacity were due to the sample itself. In
54
this technique, the reference of a lockin amplifier applies an AC voltage at frequency
f to a heater connected to the sample. The heater then produces an AC power at
angular frequency w = 47rf (the extra factor of two is due to the fact that power
is proportional to the square of the applied voltage), which in turn produces a zerofrequency rise in sample temperature (due to the fact that the applied power is always
greater than zero) and a temperature oscillation of amplitude ITac| at frequency w.
ITac is then measured by the lock-in. Due to the noise bandwidth narrowing offered
by the lockin, ITac can be measured precisely. Assuming wT > 1 and wr 2 < 1,
where 71 and 72 are as defined in Section 3.2, the heat capacity can be found from
P
C = 2wT
1
[1 + W2T+
2
22
2K]1/
W7 + 3
]-1/2
(3.5)
The prefactor 2wP
I TaI can be written QT/27r
ITac II where QT is the total heat applied over
one oscillation of the power. Hence this prefactor is the quotient of heat applied
per radian and ITacd, which is the temperature change produced over one radian.
This prefactor will then be an accurate measure of the heat capacity provided that
two conditions hold. Firstly, the time per radian 1/w must be much greater than the
response time of the sample so that the sample can follow the temperature oscillation;
i.e. wT2 < 1. Secondly, 1/w must be much less than the response time of the weak
thermal link so that no heat is dissipated to the bath over the measurement time; i.e.
wT > 1. These conditions are expressed in the second factor in (3.5). This second
factor also contains a term 2. Since K, and Kb are finite, there will be temperature
drops across both the sample and the thermal link, and a corresponding attenuation
of ITacd; this attenuation will be worse if most of the temperature drop occurs across
the sample, rather than across the link.
(3.5) can be rewritten
P
C =
D(w, T)
(3.6)
2w IT,,,
where the factor D(w, T) contains all dependence on the thermal properties of the
calorimeter. Unless the experimental conditions are such that D = 1, the ac method
will only provide a precise measurement of C/D, not of C. D
1 can occur for
two reasons. First, one could have a calorimeter with excellent thermal properties,
such that D = 1 over some range of w, but for some reason heat capacity data were
taken at some Wmeas for which D < 1 (how this might come about will become clear
in Chapter 5). In that case, if one knows ITac(wi)I at an wi for which D = 1, D =
ITac(Wmeas)I/ITac(wi)I and an accurate measurement of C can be obtained. Second,
D = 1 could occur due to poor calorimeter thermal properties, such that there would
be no w for which D = 1. In that case, D must be determined by measuring the
thermal properties of the calorimeter-including its heat capacity-using methods other
than AC calorimetry, and having an accurate model for the calorimeter that relates
those thermal properties to D. Hence, the amount of calorimeter characterization
necessary to obtain good measurements of D(w, T) makes the AC method seem much
less attractive than the relaxation method unless D = 1 can be achieved.
Whether or not D = 1 can be achieved below 1K will depend strongly on the
55
material under study. Following Sullivan and Seidel, note that the ratio of frequencyindependent to frequency-dependent terms in D(w) above is
(2LKb/3AK)/
(w 2 T22) =
6O/ (W2 L 2 Ti)
(3.7)
where q = K/pc is the diffusivity of the material. Also note that the frequencyindependent term itself is given by
2LKb/(3AK) = 2L 2 /(3rq1 )
(3.8)
In order to obtain D = 1, we want the frequency-independent term (3.8) to be
small and the ratio (3.7) to be large so that the frequency-dependent terms are even
smaller than the independent term. Hence 7r 1 /L 2 > 1 is desired. Metals typically
have q - 104 cm 2 /s below 1K. So for a typical Tr of 1s, metals will easily satisfy
this requirement with reasonable L. Electrically insulating magnetic materials such
as Sr 3 CuPt1_pIrpO 6 , however, will typically have
i1
< lcm 2 /s, since they are poor
thermal conductors with relatively large magnetic specific heats below 1K. Hence the
condition can be satisfied only by making L very small, perhaps 1mm or less.
If the sample allows D = 1 over some range of w, the AC method can provide
specific heat data of comparable accuracy and superior precision to the relaxation
method. In addition, the AC method requires less data processing and analysis, and
provides a continuous readout of heat capacity via ITac I. This latter feature allows
one to scan a thermodynamic variable such as T or H and so obtain the heat capacity
as a function of one of these variables in a relatively short time.
Even if D < 1, the AC method is still valuable to obtain a qualitative picture of a
material's specific heat. In particular, if the material has a phase transition, there will
be a peak in specific heat over a narrow temperature interval. From (3.5), this will lead
to a dip in the prefactor P/2wC, and peaks in T1 and T2 over that temperature interval.
The overall effect will be a dip in ITac. Hence when studying a new material it is
useful to use the AC method as a first step, since it will allow one to quickly identify
interesting qualitative features in the spectrum such as phase transitions. These
features can then be studied more quantitatively with the relaxation method. Finally,
one can go further if the frequency-independent term is the dominant contribution to
D over the temperature range of the phase transition. In this case, D is just a constant
scaling factor, which can be determined by measuring C at a couple of temperatures
with the relaxation method and comparing with the AC result P/2WITac. Then the
precise data taken via the ac method can be rescaled by D to give both precise and
accurate specific heat data over the phase transition.
(3.5) was derived by Sullivan and Seidel from an exact solution of the thermal
model shown in Figure 3-2. The model was solved using the matrix method [39],
in which the temperature and heat flux at z = L can be found by multiplying the
temperature and heat flux at z = 0 by a 2x2 matrix describing the thermal properties
of the material between the two points. The matrix method is easily extended to
the calorimeter geometry used for this experiment. Matrices describing the thermal
link, sample, calorimeter, and any boundary resistances between them can be easily
written down and included in the matrix multiplication. The result of this matrix
56
calculation is the "thermal transfer function", LoTacl (w), for ITacI measured at a given
position in the calorimeter.
As mentioned above, (3.5) holds provided wr 2 < 1 and wT > 1. The condition
wT2 < 1 can also be written lo > L, where L is the sample thickness and lo = L
2K§
VCW
is the characteristic thermal length (length over which T can change appreciably).
Seen in this way, this condition states that the calorimeter temperature does not vary
appreciably over the sample thickness, so that the entire calorimeter can be considered
a single thermal object with a single temperature. Under these "one-lump" conditions,
the positions of thermometer, heater, and thermal links are not important. Hence
(3.5) should apply to any calorimeter geometry, provided the one-lump criteria hold.
Hence I will refer to the Sullivan and Seidel geometry-independent (SSGI) transfer
function. One test of a well-designed calorimeter is how well it corresponds with the
SSGI transfer function having the same thermal parameters; for such a calorimeter,
the details of where and how the various elements of the calorimeter are placed are not
important. In particular, the calorimeter geometry used in this thesis will be modeled
as a single slab with two thermal link wires, one on either side of the slab. As will be
seen in Chapters 5 and 6, the transfer functions resulting from exact solution of this
two-wire model and those resulting from SSGI will differ very little.
In practice, an AC specific heat measurement begins with measurements of the
transfer function at a few points over the temperature and field range of interest, in
order to determine an appropriate frequency range for collection of specific heat data.,
Provided r/T1 /L 2 > 1, the transfer function will have a "plateau", where the transfer
function is independent of frequency. D = 1 will hold on this plateau, and specific
heat can be measured accurately for frequencies on the plateau. A frequency in this
range is chosen, and specific heat is measured as temperature or field is varied.
Since the thermal transfer functions are determined by the thermal characteristics
of the calorimeter, they can be used for thermal characterization of the calorimeter.
Also, one can measure parameters such as calorimeter heat capacity or link wire conductance via relaxation calorimetry, and use a calorimeter model to fit the transfer
function and extract other calorimeter parameters, such as calorimeter thermal conductance. Finally, if one can also estimate calorimeter thermal conductance via other
methods, one can use the model to predict the transfer function, and compare with
the measured function as a consistency check on the other measurements and on the
model. Thermal transfer functions will be used for all these purposes in this thesis
(see Chapters 5 and 6).
57
58
Chapter 4
Apparatus
4.1
Dilution Refrigerator and Magnet
Refrigeration was provided by an SHE Corporation Model 430 dilution refrigerator
that had a measured cooling power of 210pW at 100mK. Temperatures as low as
13mK at the mixing chamber were achieved with the experiment attached. Magnetic
fields were generated by an American Magnetics superconducting magnet, which provided fields up to 7T. Field uniformity was better than 4 parts in 104 over the entire
experimental area.
4.2
Heat Capacity Experiment
Because the dilution refrigerator was not top-loading, cycling even a single calorimeter
from room temperature to low temperature and back required at least two weeks. On
the other hand, the fact that the refrigerator was not top-loading also meant that
a large volume was available for experiments. This made it possible to examine
both a calorimeter with a sample and a control calorimeter without sample in a
single cooldown, and reliably separate effects due to the sample from effects due
to the calorimeter itself or to the refrigerator environment. This large volume also
made it possible to consider calorimeter designs in which the powder sample was
thin but had a large surface area. Such a design is desirable because L <
T
/pc
for good AC calorimetry, and K, = KA/L > Kb for good relaxation calorimetry.
Since K is very small for ceramics such as Sr 3 CuPtO.5 1ro.5 0 6 and electrical insulators
such as K3 Fe(CN) 6 , and since both are expected to have large heat capacity at low
temperatures, one would like to maximize A and minimize L in order to obtain good
conditions for calorimetry. These considerations provided the motivation for the
design of this heat capacity experiment, which I describe in further detail below.
4.2.1
Support Structure
The overall structure of the apparatus is shown in Figure 4-1. An OFHC copper plate
was bolted to the mixing chamber, and silver soldered to a 6. inch long, 1 inch ID/I
59
inch OD copper tube that in turn was silver soldered to another OFHC copper plate,
called the bath plate. Threaded brass rods were screwed into the bottom side of the
bath plate until a short section of rod stuck out on the top of the plate. Brass nuts
were threaded onto these short sections and tightened against the top of the plate.
This prevented the rods from rotating when other nuts were threaded onto the rod
from below, and improved thermal contact between the rods and the bath plate. A 2
inch diameter, -I inch thick copper plate was passed through the brass rods and held
in place with brass nuts. This plate provided a black body shield against radiation
from higher parts of the refrigerator.
The calorimeter stages were attached below this black body plate. A single stage is
shown in Figure 4-2. The base of each stage was a copper plate identical to the black
body plate. Screwed into the plate were three brass L brackets, with a long horizontal
vespel peg and a short vertical vespel peg glued with Stycast 2850 epoxy onto the
appropriate face of the bracket. These vespel pegs were individually machined on the
face of a vespel rod, cut off the rod, and filed to a sharp point. The calorimeter itself
consists of a mixture of powder sample and Apiezon N grease [40] sandwiched between
two 1.5 inch diameter, 2 mm thick quartz plates. The calorimeter rested on the
vertical vespel pegs, and was confined in the horizontal plane by the horizontal pegs.
Electrical leads for the calorimeter thermometers and heaters (see Section 4.2.2) were
strung through holes in the copper plate and plugged into a Microtech [41] plugboard
epoxied to the bottom of the plate. 2j mil manganin wires from this plugboard were
threaded through the copper tube to the mixing chamber, where they were connected
to another Microtech plugboard. From this plugboard, more 2- mil manganin wires,
thermally anchored to copper bobbins with GE 7031 varnish at the mixing chamber,
base plate, still, and cold plate, were used to carry electrical signals in and out of the
vacuum can.
Between calorimeter stages and between the black body plate and bath plate,
copper blocks (typical dimensions 1 inch x inch x 1 inch) were positioned to increase
thermal contact between the calorimeter stages. Once all the stages were loosely
installed on the support structure, the brass nuts on the bath plate and below each
calorimeter stage were tightened as much as possible to ensure good thermal contact
between the copper blocks and the plates.
After assembly, the entire experiment was wrapped in aluminized mylar sheet,
which was affixed to the experiment using Apiezon N grease and dental floss [42].
This shielded the experiment from black body radation emitted from the 4 K vacuum
can surfaces.
It was desired to have two four-wire thermometers and two four-wire heaters available on each calorimeter, both for diagnostic purposes and as insurance against the
likely loss of electrical contact to some devices during cooldown. Given the number of
wires available, this limited the number of calorimeters that could be measured in a
single cooldown to two (labeled a and 3). One calorimeter contained the sample, and
the other was a control calorimeter used to correct the sample data for effects due
to the addenda or the refrigerator environment. The same quantity of N grease used
for the sample calorimeter was placed between the plates of the control calorimeter,
which will be referred to hereafter as the empty calorimeter. The number of calorime60
Cu Tube
Bath Plate
Cu Block
BBR Plate
Calorimeter Stage
Threaded Brass Rod
Figure 4-1: Overall view of apparatus. Only one calorimeter stage is shown. Note
that devices (heater, thermometer) are not shown on the calorimeter.
61
CaLorimeter
Cu Plate
Vespel Peg Frame
6-32
Th ru for Brass
Rod
Figure 4-2: Wireframe view of calorimeter stage. The calorimeter sandwich (sample
dough between two quartz plates) is supported by vertical and horizontal vespel pegs.
These pegs are glued into brass L brackets.
Thermao
Link Wire
Hleaiter Wire
ThermaL Link Fingers
Chip Resistor Thermometer
Figure 4-3: Top view of one calorimeter plate, showing configuration of heater, thermometer, and thermal link.
62
ters that could be measured during a single cooldown is limited to four (assuming
one thermometer and one heater per calorimeter) by the number of electrical wires
available. Also, magnetic field profile calculations indicated that with four calorimeter stages, magnetic field variations of roughly 1% from top to bottom stage would
be expected.
4.2.2
Calorimeter
As mentioned above, the calorimeter can be described as a sandwich, with a mixture
of sample powder and Apiezon N grease sandwiched between two quartz plates. Heat
is intended to flow primarily perpendicular to the plates, so that the heat flow problem
can be considered to be one-dimensional. This design geometry takes advantage of
the large (2- inch) diameter of the vacuum can tail piece (where the experiment is
located) to minimize the sample thermal conductance. Since for this one-dimensional
case the sample conductance K, = rKA/L, a larger A not only improves K, directly,
but also makes it possible to make L smaller for a fixed sample mass. In this situation,
KS Oc #4 , where 0 is the diameter of the calorimeter plate. Given the tail piece
diameter, # can be made relatively large (1.5 inches in this case).
The material for the plates had to be chosen with care. It was desirable for it
to be an electrical insulator (since the heater and thermometer would be glued to
it), have excellent thermal conductivity, and very low specific heat. The last of these
requirements was especially important since the plate is large, and so potentially could
contribute a great deal to the overall heat capacity of the calorimeter. Of course this
is undesirable since for accurate sample heat capacity measurement, one wants the
calorimeter heat capacity to be dominated by the sample, not the addenda.
The material chosen for the plates was single crystal Z-cut quartz [43]. For temperatures below 1 K, its thermal conductivity is surpassed only by pure metals such
as copper and silver, and by superfluid 4 He. The Z-cut ensures that the 001 axis,
which has a slightly higher thermal conductivity than the other crystallographic directions, is perpendicular to the face of the plate. More impressive is the specific heat
of quartz, which below 1 K is among the lowest of commonly used low-temperature
materials [42]. Finally, it is electrically insulating. One surface of each plate was
roughened with 120 pm grit to increase the surface area of contact between sample
and plate. This was done in hopes of decreasing thermal boundary resistance.
Attached to the non-roughened face of each quartz plate were a thermal link,
thermometer, and heater. This face is shown in Figure 4-3. The thermal link consisted
of a set of copper fingers cut from a sheet of 1 mil thick copper shim stock and glued
to the plate with GE 7031 varnish, and a 21 inch long, 36 gauge copper wire affixed
to the copper fingers with silver epoxy [44]. The copper fingers were used to improve
thermal contact between the quartz plate and copper wire, and to improve the thermal
conductivity of the calorimeter in the plane of the plate below 200 mK (see Figure 45). A Dale Electronics Model RCWP-575 chip resistor (1%, 1 kQ RuO 2 thick film)was
used as a thermometer, and glued to the center of the plate with Stycast 1266 epoxy.
The heater was an 18 inch length of 1 mil diameter 92% platinum/8% tungsten wire
(called PtW wire henceforth). The wire was wound over the entire surface of the
63
plate and glued down with GE 7031 varnish. After the GE varnish dried, the thermal
link and heater were coated with a thin layer of Stycast 1266 epoxy. This was done
after it was found (in another experiment in this laboratory) that GE varnish bonds
to sapphire plates did not reliably withstand repeated thermal cycling.
The chip resistor was chosen for thermometry because of its small size, relatively
low heat capacity [45], low magnetoresistance, relative insensitivity to thermal cycling,
and its common use in our laboratory for secondary thermometry in the 0.1K to 1K
temperature range. The PtW wire was chosen because it has the lowest specific heat
of common resistance wires at low temperatures [46]. Copper was chosen for the
thermal link due to its well-known thermal properties and availability. In retrospect,
copper was not a good choice for the thermal link because of its large nuclear heat
capacity at high (several tesla) fields and low temperatures. Also, the use of industrial
copper shim, which likely has large quantities of unknown impurities, was not wise.
High purity silver would have been a much better choice.
In the K 3 Fe(CN) 6 experiment and the initial Sr 3 CuPt. 5 Iro. 5O 6 experiment, 5 mil
diameter NbTi wires were used as electrical leads connecting the thermometers and
heaters to the Microtech plugboard. NbTi wires are superconducting in fields of 8T
below 1K, and so have excellent electrical conductivity but very low thermal conductivity and specific heat. These are excellent properties for electrical leads for lowtemperature calorimetry. On the other hand, these wires are not at all malleable and
so often could not be plugged in without placing them under stress; furthermore, it
is very difficult to make reliable solder joints with them. As a result, one of the four
calorimeter thermometers was lost during cooldown of the K3 Fe(CN)6 experiment,
and three of four were lost during the first cooldown of the Sr 3 CuPtO.5 Iro.0 O 6 experiment. For the second Sr 3 CuPt0 .5 Ir 0 .5 0 6 experiment, the NbTi wires were replaced
with 1 mil PtW wires, which are thin enough to have thermal properties comparable
to those of NbTi, but are quite malleable, solderable, and surprisingly robust (1 mil
diameter manganin wires, for example, are much more fragile).
With the calorimeters mounted on their respective stages, the copper thermal
link wires were threaded through holes in the copper plates to the bath plate, where
they were placed between one of the copper blocks and the bath plate before final
tightening of the brass nuts. When the brass nuts were tightened, the thermal link
wires were pressed firmly between blocks and plate, ensuring good thermal contact
to the bath plate and hence to the mixing chamber of the refrigerator.
4.2.3
Calorimeter Models
Models for the expected behavior of the empty and K3 Fe(CN) 6 calorimeters were
developed for comparison with experimental data. In these models it was always
assumed that the calorimeter could be modeled as a one-dimensional system; that is,
the calorimeter temperature T was a function of z only, where the z axis is perpendicular to the plane of the calorimeter plates. This assumption is reasonable because the
heater wire and thermal link fingers are distributed uniformly over the entire surface
of the plates, and because the thermal conductivity of quartz is excellent.
The expected heat capacities of various components of the empty calorimeter are
64
shown in Figure 4-4. The expected thermal conductances of various components
of the calorimeter, and thermal boundary resistances between various surfaces, are
shown in Figure 4-5. These data are based on published specific heat and thermal
conductivity data for the various materials used, and on measurements or estimates
of the relevant geometrical factors. In Chapters 5 and 6, these expected values will
be compared with values of link conductance and calorimeter heat capacity measured
with the relaxation method.
10 3
OkG
mt cal, nea'su'red'
total calcd addenda
zero field
*CPI
N grease
Quartz
Cu fingers
46 l1d) Dale Chip
PtW htr/Iead wires
Cu wire
104
O
0
0
0
-
0
V
*
0
*
L~3
VO
0- 6
dt
0
y,
0
0
00
V
A AA
b~
A
A,
A,
Cj
0
0 0
a NU
A A
10~~7
V
10-8
0.0 1
I
0.02
0.05
0.1
I
0.2
0.5
1
2
5
10
Temperature [K]
Figure 4-4: Expected heat capacity of various calorimeter components. The N grease
data below 0.4 K was obtained by linear extrapolation, which is reasonable for glassy
materials [47]. The quartz data below 0.3 K is a T 3 extrapolation, which underestimates the true heat capacity [47]. Also shown are expected empty calorimeter heat
capacity and measured empty calorimeter heat capacity from Sr 3 CuPtO.5 1ro. 5 0 6 experiment, and Sr 3 CuPtO.5 Iro.50 6 heat capacity measured here (below 1K) and by
Ramirez [15](above 1K).
Also, thermal transfer functions for the empty calorimeter were computed using
these expected values. The calorimeter was modeled as a single slab of material with
a thermal link wire on either side. I will refer to this as the two-wire model. The
matrix method was then used to derive exact, analytical expressions for the thermal transfer functions. These transfer functions will be compared with experimental
transfer functions in Chapters 5 and 6.
65
10 2
I
I I . . I, . I
I
I
.
I .
. I I I
VV VV
10 0
V
V
V
v Quartz normal
Quartz/sample bdry
---Cu finger/quartz bdry0 Quartz in plane
A N grease
4 Cu finger
o Copper wire
_
* vespel pegs
* PtW wires
* NbTi wires
-
V
102
V
0000
C-U
C
0
~4-)
Ui
-4
D~
T
III
Ile
10
V
C
0
CdVO3
do
00
0
0
0
000000
OC
0 000000
IR 10L
CU
*
0-
00
100
0.
01
0.02
WV
0.05
0.1
0.2
0.5
1
2
5
10
Temperature [K]
Figure 4-5: Expected thermal conductance of various calorimeter components. The
PtW entry is for 16 1 mil diameter PtW wires, each 1 inch long (representing the
heater and thermometer lead wires). The NbTi entry is for the same number and
lengths of 5 mil diameter NbTiwires. The quartz/sample boundary resistance is taken
to be the same as that between copper and glue.
66
4.2.4
Thermometry
Primary thermometry for 0.1K < T < 0.3K was provided by a 3 He melting curve
thermometer (MCT) of the type described by Greywall and Busch [48]. The MCT is
thermally anchored to the mixing chamber, and its capacitance measured by a tunnel diode oscillator located on the coldplate. The accuracy of temperatures measured
with the MCT was typically better than +lmK. As mentioned above, secondary thermometry was provided by Dale chip resistor thermometers. For the K3 Fe(CN)6 experiment, the mixing chamber chip resistor was calibrated against the MCT, and the
calorimeter chips calibrated against the mixing chamber chip with the mixing chamber temperature controlled. For the Sr 3 CuPtO.5 1ro. 5 0 6 experiment, resistances of chip
resistor thermometers on the calorimeters and mixing chamber were recorded along
with the corresponding melting curve thermometer temperature as the refrigerator
was slowly cooled. The two methods did not give significantly different results. For
the /3 calorimeter top plate thermometer, the maximum difference between the two
methods was 6mK at 120mK, decreasing to a negligible amount by 209mK; for the
o calorimeter top plate thermometer, the maximum difference was 2 mK at 133 mK,
and was less than lmK at lower and higher temperatures.
Magnetoresistance was measured by temperature controlling the mixing chamber and measuring chip resistance as a function of magnetic field. The resulting chip resistance versus temperature and field data were fit to R(T) = (A +
RHH) exp(B/Ti) [49].
For the four calorimeter thermometers, typical values are
1
A = 0.37kM, B
1.67K4, and RH = 2.6Q/T. Variations in these parameters between thermometers are typically 5%. Chip resistor temperatures are typically taken
to have errors of ± 2mK, due to scatter about the fit. For temperatures above 0.3K,
the R(T) fit was extrapolated.
It was assumed that the chip resistors and MCT were in excellent thermal contact, and so had no appreciable temperature difference, provided that R(T) fit well
to the expression given above for the chip resistor. It has been shown [49] that even
under conditions of excellent thermal contact, the chip resistor begins to deviate from
the exp(B/Ti) behavior below 50mK, and that the behavior below 50mK depends
strongly on the device. However, it was found during the second Sr 3 CuPtO.5 1ro.5 0 6 ex-
periment that all of the chips began to deviate at higher temperatures. For the empty
calorimeter top thermometer, deviation began around 60mK; for the empty calorimeter bottom thermometer, 80mK; and for both of the Sr 3 CuPtO.5 Iro.5 0 6 calorimeter
thermometers, around 120mK. Heat capacity data was not taken below the deviation
temperatures, or was taken only to get a qualitative idea of the heat capacity below
those temperatures.
Since all of the thermometers were from the same batch, it is likely that this difference between thermometers is caused by loss of excellent thermal contact between
each thermometer and the MCT. For the empty calorimeter thermometers, it is likely
that the bottom thermometer is not as well contacted to the quartz plate as the
top thermometer. For the Sr 3 CuPtO.5 1ro. 5 0 6 calorimeter thermometers, it is possible
that neither thermometer is as well contacted to the quartz plates as is the empty
67
calorimeter top thermometer. It is also possible that the thermal link wires for the
Sr 3 CuPtO.5 1r o.5 0 6 calorimeter are not as well-connected to the bath plate as those of
the empty calorimeter.
Finally, it is important to address the question of the lowest possible temperature
at which heat capacity could be measured with the current apparatus. The MCT has
measured temperatures as low as 13mK at the mixing chamber. Also, the calorimeters
did continue to cool below the deviation temperatures (as evidenced by continued
increase in resistance). Assuming that these thermometers have similar calibrations to
others used in our laboratory, it is found that the empty calorimeter top thermometer
has measured temperatures as low as 20mK, while all other thermometers (including
the chip on the mixing chamber!) stop cooling in the 40 to 50mK range. Hence
with more careful attention to thermal contact issues, it seems possible to cool the
calorimeters to 20mK. This does not necessarily imply that heat capacity could be
measured to that temperature. For the AC method, there is always a DC component
of the power, proportional to the peak AC power, which will warm the calorimeter
above the bath temperature during measurements. This can be reduced, but only
to the extent that the signal-to-noise of lTac is not too low. Also, the resolution of
the relaxation method is limited by the size of the throw AT(0) required to achieve
reasonable signal-to-noise. However, these issues of thermal contact and signal-tonoise are technical, and in principle solvable, problems. Therefore it seems feasible,
with some effort, to extend the measurement range of this apparatus to at least 50mK,
and perhaps lower.
68
Chapter 5
Potassium Ferricyanide
Experiment
The specific heat of potassium ferricyanide K 3Fe(CN) 6 was measured as a test of the
apparatus. This material (Figure 5-1) contains chains of Fe3+ ions, with effective
spin-half and nearest-neighbor coupling. The chain axis is parallel to a. The coupling
between nearest neighbors in the chain contains an isotropic part J of strength J ~
0.36K, and an anisotropic part, with J, ~ 0.1K, Jb
Ja ~ -0.03K
(as usual,
J > 0 indicates AF coupling). The coupling Jd between chains is isotropic and much
weaker, Jd ~ 0.01K [50]. When cooled below 1K in zero field, K3 Fe(CN) 6 develops
short-range antiferromagnetic order along the chains, with the spins in a given chain
aligning antiparallel to one another and pointing along the c axis. The short-range
order is described approximately by the 1D Ising model [50], and leads to a broad
plateau in the specific heat between 200mK and 500mK [51]. K3 Fe(CN) 6 undergoes
a transition to antiferromagnetic long-range order at TN = 129 mK. The transition
to long-range order is observed as a peak in specific heat at 129 mK. (Note that the
transition to long-range order occurs at higher temperature than would be suggested
by Jd; this is due to uncertainties in Jd and to short-range order in the chains above
TN, as discussed in Section 2.1.) Both the short- and long-range order are strongly
affected by magnetic fields as low as 100 gauss [52].
This material was chosen to test the apparatus for three reasons. First, the sharp
feature at 129 mK has been observed by multiple groups and so provides a test of
temperature and heat capacity measurements. Second, since the specific heat has a
strong response to weak (hundreds of gauss) applied fields, it provides a test of heat
capacity measurement in a field. Finally, K3 Fe(CN)6 is electrically insulating and
has a quasi-one-dimensional structure, so it should have similar thermal properties to
Sr 3 CuPtO.5 1ro.5 0 6 .
5.1
Calorimeter Preparation
For this experiment, calorimeter preparation began with two complete, previously
unused calorimeter plates. 6.44mg of small K3 Fe(CN)6 crystals (Fluka, > 99%,
69
I0
IO
I
I
'
I
O
bN
I
Figure 5-1: Monoclinic unit cell of K3 Fe(CN) 6 . Closed spheres represent Fe, open
spheres K, open diamonds C, and closed squares N. 3, the angle between a and c, is
approximately 1070, and a = 7.04A, b = 10.44A, and c = 8.4A. Six cyanide groups
surround each Fe, in approximately octahedral coordination. Here, only two Fe ions
are shown with all cyanide groups. The closest cyanide groups on nearest neighbor
sites are 2.74A apart on the chain axis a. For nearest neighbors along b, the closest
cyanide groups are 6.14A apart.
70
stock no. 60300) were measured out using a Mettler balance. 13.7mg of Apiezon
N grease, measured with the same balance, was placed in an agate mortar with the
K3 Fe(CN) 6 crystals. The two components were ground and mixed with an agate
pestle until a reddish-orange "dough" was obtained. The dough was carefully and
thoroughly scraped out of the mortar and onto one of the calorimeter plates. The
second plate was pressed against the dough to form the calorimeter sandwich. The
sandwich was placed in a hobby vise and the dough compressed further, taking care
not to lose any dough out the sides of the sandwich. The completed calorimeter was
mounted on the vespel peg brackets, and the NbTi leads plugged into the microtech
connector under the stage plate.
The empty calorimeter was assembled similarly. Here, 68mg of Apiezon N grease
was used. Some grease was lost out the sides of the calorimeter during compression;
the final amount of grease in the calorimeter was determined by massing the calorimeter before and after assembly. The amount of grease on this calorimeter was originally
intended to correspond with the amount used to make K3 Fe(CN) 6 dough on an earlier
run, in which far too much K3 Fe(CN)6 was used (leading to extremely long thermal
time constants). When an appropriate amount of K3 Fe(CN) 6 (6.44 mg) was finally
used, it was decided not to reduce the amount of N grease on the empty calorimeter to
13.7 mg due to the expectation that the addenda (calorimeter and grease) would have
a negligible contribution to the total heat capacity of the K3 Fe(CN) 6 calorimeter, and
the fragility of the calorimeter.
The empty calorimeter stage was mounted in the top position of the support
structure, with the K3 Fe(CN) 6 calorimeter stage below it. The entire experiment was
then bolted to the mixing chamber of the refrigerator.
5.2
AC Method Procedures
Thermal transfer functions were measured at a few temperatures over the temperature
range of interest, 100mK to 450mK. As discussed in Chapter 3, thermal transfer
functions are needed to determine the range of frequencies that will afford D = 1
over the temperature or field range of interest, and are generally useful for thermal
characterization and modeling of the calorimeter. Typically, the excitation power
angular frequency (EPAF) range used was 0.126 rad/s to 16.08 rad/s. (EPAF is w
from Section 3.3).
The setup diagrammed in Figure 5-2 was used for transfer function and AC specific
heat measurements. Mixing chamber temperature control was achieved with a homemade AC resistance bridge and homemade lockin operating at 1 kHz to measure mixing chamber chip thermometer resistance, and a homemade PID temperature control
circuit to drive the mixing chamber heater. For the K3 Fe(CN) 6 (empty) calorimeter,
an AC voltage was applied to the calorimeter bottom (top) heater with the reference of a Stanford Research Systems SR830 lockin amplifier, and the calorimeter top
(bottom) thermometer resistance measured by a Linear Research LR-400 Four-Wire
AC Resistance Bridge in lOAR mode. (For the K3 Fe(CN) 6 calorimeter, the top thermometer was used because the bottom thermometer wiring failed during cooldown.
71
IMixing Chamber
LR-400
AC Resistanc e
Bridge
RB
x1O AR
Hlomemade
Tiemperature
Controller
output
Input
Excitation
Lockin
PC
Linux Box
Figure 5-2: Apparatus used to measure AC heat capacity for Sr 3 CuPtO.5 IrO. 5 0 6 and
K 3 Fe(CN) 6 experiment.
72
The bottom heater was used to to ensure that heat went through the sample before
reaching the thermometer. For the empty calorimeter, the bottom heater wiring failed
during cooldown, so the top heater and bottom thermometer were used.) The LR-400
reading was fed to the SR830, which detected the RMS amplitude IVUcJ and phase #
(at twice the frequency of the excitation voltage). A PC running Linux directed the
SR830 (via GPIB) to acquire 75 amplitude and phase readings on this signal over five
to ten lockin time constants (30 seconds here), and to report them to the PC.
The reported ITac was found by first taking the mean and standard deviation
of the 75 IVaCI readings and converting them to temperatures using the MCT calibrations. At least two additional corrections were made. First, the lockin reports
the RMS value of IVacI, while the Sullivan and Seidel formula uses the amplitude
of ITacd; to account for this, all measured IVaci were multiplied by v/2. Second, the
electronics used for ITac measurement have their own transfer functions. The heater,
thermometer, and associated wiring did not produce any attenuation of signals over
this frequency range. However, the LR-400 bridge significantly attenuated signals
above 0.2 Hz. The bridge transfer function was measured with a JFET (Motorola
2N5486) operated as a variable resistor [53]. A small (6 mV) signal from the lockin
reference was applied to the gate of the JFET, and the drain-source resistance of the
device (which oscillated at the same frequency as the gate voltage) measured with
the LR-400/SR830 setup described above. Figure 5-3 shows the measured LR-400
transfer function. Given this transfer function, IVac at a given w was corrected for
LR-400 attenuation by multiplication by the appropriate factor. Note that for a given
frequency f in the figure, the corresponding EPAF is 27rf (as opposed to 47rf), since
the LR-400 detects ITac directly.
Unfortunately, the effects of the LR-400 transfer function were not discovered until
after the experiment was over. Hence the thermal transfer functions used to determine an appropriate frequency for specific heat measurements all had a false knee at
high frequency due to the LR-400 response. To clarify this point, the uncorrected
and corrected transfer function for the empty calorimeter at 102 mK are shown in
Figure 5-4. For the uncorrected function, there is a narrow "plateau" centered at
1.89 rad/s, which was chosen as the EPAF for the heat capacity experiments. Recall from Chapter 3 that in order to obtain accurate heat capacity data from the
formula C = P/2wITacI, one must choose w on the "plateau" of the transfer function. Otherwise, one must know ITacI (Wpat), that is the value of ITacl for some Wplat
that is on the plateau of the transfer function at the given temperature. This is
needed in order to determine the off-plateau correction factor D(w), which will equal
ITac (Wmeas) / Tac I(wpat) , where Wmeas is the EPAF at which heat capacity was actually
measured.
For the situation shown in Figure 5-4, the chosen EPAF of 1.89 rad/s apparently
is below the plateau (not shown in Figure 5-4) of the actual (corrected) function.
Hence to obtain accurate heat capacity data, Tac(wpiat) has to be known so that
D(w) can be determined. For the empty calorimeter, the EPAFs chosen (1.76 rad/s
or 1.89 rad/s) based on the uncorrected transfer functions were well below plateau;
moreover, the plateau region indicated by the corrected thermal transfer functions
began at frequencies above 12.57 rad/s, above the region of frequencies examined for
73
100
50
20 10 -0
E
0
gos
.
100
5
2 10.5
0
0.2
0
0.1
0.01
0.02
0.05
0.1
0.2
0.5
1
2
5
10
Frequency [Hz]
Figure 5-3: LR-400 bridge transfer function measured with JFET.
the transfer function and in a region where attenuation due to the LR-400 was severe
(factor of 19 at 1.28 Hz, corresponding to EPAF of 8.04 rad/s). Hence Tac(wpiat) cannot be determined, and it is only possible to partially correct the empty calorimeter
heat capacity data for this off-plateau choice of Wmeas (Section 5.4). In particular,
only an upper limit can be placed on the empty calorimeter heat capacity. For the
K3 Fe(CN) 6 calorimeter, no off-plateau correction had to be applied, since the EPAFs
chosen (0.377 rad/s or 0.503 rad/s) were on plateau for both the uncorrected and corrected transfer functions. With an excitation voltage frequency determined, ITac data
were acquired, using the procedure described above, as a function of temperature at
fixed field, with the magnet running in persistent mode. For the K3 Fe(CN)6 calorimeter, several fields from zero to 9.5 kG were examined. For the empty calorimeter, only
zero field was examined.
5.3
Thermal Relaxation Method Procedures
In order to measure heat capacity via the relaxation method, it is necessary to measure
Kb and the decay of calorimeter temperature with time AT(t) upon turning off the
calorimeter heater (see Section 3.2). In addition to providing the heat capacity,
these data are useful for thermal characterization of the calorimeter. Obviously, the
direct measurement of Kb is useful for thermal characterization. Also, the number of
exponential decays needed to describe AT(t), and the fit parameters of each, can be
used to extract information about calorimeter thermal properties.
74
1 06
S
50
*
0
Corrected
o Uncorrected
0
2
C,)
0
L
5
105
0~0
35
0
0
2
10 4
0.1
0.2
0.5
1
5
o [rad/s]
2
10
50
20
100
Figure 5-4: Effect of LR-400 transfer function on empty calorimeter 102 mK thermal
transfer function. The transfer function is normalized for power.
Excellent temperature control of the heat bath is essential. Several minutes are
required to measure Kb and AT(t), and fluctuations in heat bath temperature over
that time will produce errors in Kb and AT(t). In this experiment, the heat bath is
the mixing chamber of the refrigerator. Bath temperature control was achieved in
the following way. A voltage proportional to the mixing chamber chip resistance was
measured with a homemade ac resistance bridge and lockin amplifier. This voltage
was input to a Princeton Applied Research Corporation Model 113 pre-amplifier,
where it was compared with a setpoint voltage. The difference between the two
voltages was amplified and sent to a homemade PID temperature control circuit,
which drove the mixing chamber heater. This setup provided ±10Q stability of the
mixing chamber chip resistance at 4 kQ, or +2 mK at 239 mK. As will be seen, even
this degree of stability led to substantial errors in C.
With temperature control established, the power Po to be applied to the heater
for the AT(t) measurement was determined. PO was chosen to be large enough so
that the signal to noise in the AT(t) data was at least 10, but small enough so that
the temperature resolution of the resulting heat capacity data was good. For the
K 3 Fe(CN) 6 experiment, this led to PO's that produced AT/T
-
3%.
Given PO, the next step was to measure Kb. Kb can be determined from
P = Kb AT(P)
(5.1)
where P is power applied to the calorimeter bottom heater, and AT(P) the resulting
75
top thermometer temperature change. Various powers P less than or equal to P
are applied to the heater using a homemade current source, and the resulting ATs
recorded. Each data point was acquired after waiting 7 to 10 link time constants.
Initially, the heater voltage was incremented in equal steps to a maximum voltage,
then decremented back to zero voltage to check for non-equilibrium effects. It was
found later that the baseline temperature drifted during the measurement. In subsequent measurements, the temperature baseline (zero power) was checked between
every two or three power/temperature points in order to correct for this. The powertemperature curves are then fit to a line, and the slope is taken to be Kb. Representative power-temperature curves obtained for the link conductance measurements
are shown in Figures 5-5 and 5-6.
4.5
3.0
01.5
* Data
- - Fit
0
0
-
117.5
118.0
118.5
119.5
119.0
Temperature [x10
120.0
120.5
3 K]
Figure 5-5: Typical power/temperature curve for K 3 Fe(CN) 6 calorimeter, without
baseline drift correction.
Finally, AT(t) was measured. The low bandwidth of the LR-400 bridge (see Figure
5-3) would be expected to distort AT(t) with time constants of a few seconds or less.
However, typical time constants found in AT(t) measurements were in the range 10
to 60 seconds for the K3 Fe(CN) 6 calorimeter, so the LR-400 was deemed adequate for
measurements of AT(t) on this calorimeter. AT(t) was measured simply by recording
the lOAR output of the LR-400 bridge as a function of time with a digital voltmeter
(Hewlett Packard Model hp34401a). These data were then transferred to the PC for
further analysis.
This method did not give accurate results for the empty calorimeter, where time
constants of less than 1 second were encountered when measured with this method.
However, the method gives an upper limit on the time constants for the empty
76
II
'I
'I
4.5
-3.0
1.5 -
0.238
--
0.240
0.242
Temperature [K]
Data
Fit
0.244
0.246
Figure 5-6: Typical power/temperature curve for K 3 Fe(CN) 6 calorimeter, with base-
line drift correction.
calorimeter. The heat capacity of the empty calorimeter calculated from this upper limit was so small compared to that of the K3 Fe(CN) 6 calorimeter that the
empty calorimeter had no effect on the K3 Fe(CN) 6 results, to within experimental error. Hence no further improvements to the method were necessary to determine
the K3 Fe(CN) 6 heat capacity.
5.4
Empty Calorimeter Results
In Figure 5-7, thermal transfer functions for the empty calorimeter at 102 mK,
279 mK, and 430 mK are shown, normalized for the applied power. Clearly the
chosen EPAFs of 1.76 rad/s and 1.88 Hz are below plateau for all temperatures.
As mentioned in Section 5.2 above, this is because the effects of the LR-400 bridge
transfer function were not recognized and corrected for until well after the data was
taken. The thermal parameters of all materials in the empty calorimeter can be computed from published data, and plugged into a calorimeter model to obtain expected
transfer functions for comparison with the measured data. The two-wire model introduced in Chapter 3 was used. The parameters for the model were determined as
follows. Since the thermal conductivity of N grease is 9 orders of magnitude less than
that of quartz, the effect of the quartz on slab conductance was ignored and slab
conductance computed using interpolated (or for T < 0.4 K, extrapolated assuming
C/T = constant) values of N grease thermal conductivity. The dough surface area
77
106
5
2
1: 5
0,
C'o,
_-
2
102mK Measured
.0
a 0-o.-
E"
-102mK
a 279mK
279mK
* 430mK
430mK
4
Materials data
Measured
Materials data
Measured
Materials data
2
,
,
10
0.1
0.2
0.5
1
2
5
10
20
50
100
w [rad/s]
Figure 5-7: Empty calorimeter thermal transfer functions, measured and as calculated
using the two-wire model and published data on calorimeter materials.
was taken to be the surface area of the calorimeter plate, and dough thickness was
determined to be 5 - 10 mil from rough measurements using a light microscope and
reticle. The thermal link wire conductance was determined using the commercial
copper wire data of Suomi [54]. Slab heat capacity was the sum of that of the two
quartz plates, 68 mg N grease, two copper fingers, two chip resistors, and two PtW
wire heaters. The amplitude of the heater voltage was as measured with an hp34401a
digital multimeter on the voltage leads of the heater. The heater resistance was as
measured with the LR-400 bridge at 4 K.
Given these input parameters, the expected transfer functions were computed
(with no adjustable parameters) and compared with the measured functions for
102 mK, 279 mK, and 430 mK (Figure 5-7). Given that the properties of materials used here could differ from those used for the published data by as much as a
factor of two, and the error in the dough thickness measurement, the agreement is
relatively good. The fact that the modeled functions fall off faster than the measured
ones as frequency decreases is due in part to the fact that the expected Kb is substantially higher than was measured. Since the plateau and high frequency knee are
not present in the measured functions, it is only possible to put a (very weak) lower
limit on the sample conductance. Comparing expected transfer functions for various
K, with the data, it is found that K, > 5pW/K for all temperatures.
AC C data for the empty calorimeter are shown in Figure 5-8. These data were
multiplied by the factor ITacI(W = Wmax)/TacI(W = Wmeas), where Wmax was the max78
imum power angular frequency at which the transfer function was measured, and
Wmeas was the angular frequency at which C/D was measured.
From Figure 5-7,
this factor was taken to be five. Hence these data represent an upper limit on the
heat capacity of the empty calorimeter. The points are all at least a factor of thirty
below the K3 Fe(CN) 6 data, which confirms that it is safe to ignore the addenda in
computing the K3 Fe(CN) 6 heat capacity.
1.0 -
0.6
* AC top therm,bot htr _
o AC top therm,top htr
*
0
0.8 -
*
-
00
.0
,X4 E
0.2
H
0
0.05
0.20
0.35
0.50
Temperature [K]
Figure 5-8: Upper limit on AC empty calorimeter heat capacity.
5.5
K 3 Fe(CN) 6 Calorimeter Results
For the K3 Fe(CN) 6 calorimeter, thermal transfer functions were measured at 108mK,
139mK, and 250mK. These are shown in Figures 5-9, 5-10, and 5-11. Clearly the
chosen EPAFs of 0.38 rad/s and 0.50 rad/s used for specific heat measurement are
on plateau for all temperatures. The plateaus are quite broad, spanning at least one
order of magnitude in w. Indeed, the low-frequency knee is below the measured range
of frequencies for the two low temperature functions, and the high-frequency knee
is above the measured range for the 250 mK function. Given this broad plateau,
one might expect that D = 1 will hold for the K3 Fe(CN) 6 calorimeter, and that the
AC specific heat data will be accurate as well as precise. The plateaus are at lower
frequencies than those for the empty calorimeter. This is not surprising, since C is
much larger for the K3 Fe(CN) 6 than for the empty calorimeter, since Kb and K, are
expected to be comparable for the two, and since we have w, oc Kb/C and wh oc K/C
for the locations w, and wh of the low- and high-frequency knees.
79
103
-
* Measured
2 Wire Model, Exact
SSGI
2
3)
04
0 -
3
2
10-5 1
0.0 1
0.02
0.05
0.1
0.2
0.5
w [rad/s]
1
2
5
10
Figure 5-9: K 3 Fe(CN) 6 calorimeter 108 mK thermal transfer function, with fit to
two-wire model and resulting SSGI transfer function.
10-4
9
8
7
6
5
4
En
-c__
3
2
* Measured
-2
Wire Model, Exact
-- SSGI
10-5 1
0.0 1
0.02
0.05
0.1
0.5
o [rad/s]
0.2
1
2
5
10
Figure 5-10: K 3 Fe(CN) 6 calorimeter 140 mK thermal transfer function, with fit to
two-wire model and resulting SSGI transfer function.
80
10
1
9
-2
5
...
1
Measured
Wire Model, Exact
SSGI
2
S4
L-z 10-
35
2
10-5,
0.01
0.02
0.05
0.1
0.2
0.5
1
2
5
10
o [rad/s]
Figure 5-11: K3 Fe(CN)6 calorimeter 303 mK thermal transfer function, with fit to
two-wire model and resulting SSGI transfer function.
Since the thermal conductance of the K3 Fe(CN) 6 dough could not be found a priori
from published data, no attempt was made to compare the measured transfer functions with what would be predicted given this calorimeter geometry and published
data on the materials involved. However, the data were fit to the two-wire model with
one adjustable parameter, the calorimeter thermal conductance. For these fits, the
error bars on the points initially were taken to be the standard deviation of the JTad
data, and generally were smaller than the plotting symbols. The scatter is generally
greater than the error bars, particularly at high frequencies. The cause for this scatter
in the transfer functions is not known. The effect of the scatter and small error bars
was that fits of the data to calorimeter models gave unreasonably large x2 . Given
this, estimates of calorimeter thermal conductance were extracted from the data in
two ways. First, the data were fit to the model with equal weight assigned to each
point, rather than weighting with 1/o.2 as is usual for x 2 fitting. Second, various
two-wire transfer functions assuming specific values of K, were calculated and compared with the data. If the assumed K, was too small, the calculated curve showed
a high-frequency rolloff not seen in the data. From this analysis, a (very weak) lower
limit on K, was obtained.
The parameters for the fit were determined as follows. The thermal conductance
of a single link wire was taken to be half the slope of the power/temperature curve,
interpolated linearly from power/temperature data taken at nearby temperatures for
the relaxation method. The dough surface area, heater voltage, and heater resistance
were measured as for the empty calorimeter functions. Finally, for the 108 mK and
81
Temp
(mK)
108
140
303
2WTF
Fit
(pW/K)
40 t 8
55 t11
85 + 20
2WTF
Low. Lim.
(pW/K)
20
30
40
Kb/Ks
Range
0.02-0.05
0.03-0.07
0.07-0.18
Table 5.1: K3 Fe(CN)6 calorimeter K, estimates. K, was determined from a fit of
the transfer function to the two-wire model (2WTF Fit), and a lower limit on K,
was determined by comparison of the data and calculated two-wire transfer functions
with various K, (TF Low. Lim.).
139 mK functions, the slab heat capacity was taken to be the value found via AC
specific heat measurement, which corresponds with that published by Fritz et al. [52]
for 6.44 mg K 3 Fe(CN)6 single crystals (the correction for the 13.7 mg N grease present
was three orders of magnitude below this and so was ignored). (Note that the data
reported by Fritz was actually compiled from measurements by Rayl et al. [51] and
by Domb and Miedema [55]. However, I will refer to this as the Fritz data.) For
the 250 mK function, the heat capacity measured with the relaxation method was
used. The heat capacity values were taken in this way because the heat capacity
of K3 Fe(CN) 6 is a very steep function of temperature below 250 mK, so even slight
differences (10 mK) in the temperatures at which relaxation and transfer function data
were taken were enough to make the relaxation heat capacity data inappropriate for
transfer function modeling.
Given these input parameters, the functions were fit to the two-wire model as
described above, with the calorimeter thermal conductance K, as the adjustable parameter. The resulting K, values determined for various temperatures are shown in
Table 5.1. The table also shows the lower limits on K, obtained by comparing calculated two-wire transfer functions with the data. The thermal parameters obtained
from the two-wire model were used to obtain the SSGI transfer function (see Chapter
3). The two-wire and SSGI model functions are shown in Figures 5-9, 5-10, and 5-11
above. The models correspond well with each other and with the measured data on
the plateau region, as expected. Small deviations are observed at the high-frequency
knee. At high frequencies, the internal structure of the calorimeter will be probed
most, so one expects the exact locations of heater and thermometer to begin to play
a role, and for deviations to therefore occur between the SSGI function and the exact
model. The fitted values for K, confirm that Kb/K is small, and our expectation
D = 1 on the plateau to within a few percent. Hence it is expected that the AC
specific heat data will be accurate as well as precise. Finally, again since Kb/K, is
small, and since the two-wire and SSGI models fit well, the relaxation AT(t) should
fit to a single time constant.
Zero field AC C/D(w) data for the K 3 Fe(CN)6 calorimeter is shown in Figure
5-12. For comparison, the Fritz and relaxation data are also shown. The AC data
82
were collected at various times over the course of two months, during which time the
refrigerator was warmed to 2K several times, and to 77K once. The data show excellent precision, notwithstanding the (gentle) thermal cycling to which the calorimeter
was subjected. Precision on the order of 2% is typical. The relationship of these data
to the Fritz data will be discussed further below. AC data was also taken at various
fields below 1 T (Figure 5-13). As expected, the peak is suppressed by application of
even a low field [52].
1.6
* AC Data, OkG
o Relaxation data, OkG
o Fritz Data, 6.44mg+Ngrease
1.4
1.2
>1.0.
Z0.8 -
*
x
U
14
0.6 -
0
to
0
I
-
0.4 -e
0.2 0
0.2
0.1
0.4
0.3
Temperature [K]
0.5
Figure 5-12: K3 Fe(CN) 6 calorimeter zero field AC and relaxation heat capacities,
with data for K 3Fe(CN) 6 single crystals published by Fritz.
The thermal relaxation method (cf. Section 3.2) was used to obtain calorimeterto-bath link conductances and to measure zero-field heat capacity for K3 Fe(CN) 6 at
four temperatures between lOOmK and 300mK. As discussed in Section 3.2, this data
is useful to obtain accurate heat capacity measurements when the thermal properties
of a calorimeter are not ideal for accurate AC heat capacity measurements, and to
provide an accurate measure of the link conductance for modeling of calorimeter
behavior.
To check for possible 2 effect, the decay curves were fit to a sum of two exponentials. These fits gave equal time constants for the two exponentials, and split the
throw evenly between them. This indicates that a single exponential is sufficient to
describe the decay, and that there is no observable 2 effect. Since heater currents
could be measured to ± 0.1 pA, and the heater resistance was known to better than
± 2 Q, yielding errors in power of typically ±0.4 nW at 100 mK and ±1 nW at
250 mK. With these errors, X2 near one were typically obtained for the linear fits
T
T
83
1.6
AC
* AC
A AC
AC
1.4
Data,
Data,
Data,
Data,
OkG
0.8kG
7kG
9.5kG
1.2
,'1.0
0
-
e0
00
C)
0
.0.8
02
%.)
0.6
A
A
.{A0
-
*
0
0.4
0.2
*
0.1
l
0.2
0.3
Temperature [K]
0.4
I
0.5
Figure 5-13: K 3 Fe(CN)6 calorimeter field-dependent AC heat capacity.
of the power/temperature curves. At a given temperature, AT(t) and Kb were measured multiple times, and the resulting T and Kb values averaged to yield final values
T and Kb. The errors in i and Rb were taken to be the standard deviations of the
distributions of measured T and Kb when these were measured more than twice, and
taken to be half the difference between the measured values when measured twice.
The calorimeter heat capacity was then calculated as C = Kbr. No correction was
applied for the addenda heat capacity, since the addenda contribution was smaller
than the error bars (see Table 5.2).
The resulting heat capacity data are shown in Table 5.2, along with the Fritz
data. The "errors" in the Fritz data are not due to errors in their data, but to
my uncertainty in the temperature, which in turn is a result of the temperature
resolution of the relaxation method used to take my data. Obviously, the errors
in the relaxation data are large. The main sources of error are the consequence of
inadequate temperature control over time scales of several minutes. One source of
error was that all of the Kb points, with the exception of two taken for the 245 ± 5mK
point, were taken without correcting for drift in the baseline temperature, which
occurred over the Kb measurement time of several minutes. This led to errors in Kb
in the range 10% to 20%. Another source of error was inadequate temperature control
during AT(t) measurement, in which typical time constants were 10s above the peak
and 30 - 50s below and near the peak, and a single AT(t) measurement included ten
time constants or more. This led to T errors at the 10% level, although at higher
temperatures (200 - 250mK) temperature stability was better and
84
T
errors were at
Temp
(mK)
95 ±5
131± 4
210
Empty Cal.
C
(pJ/K)
1.3 0.3
-
5
245 + 5
K3 Fe(CN) 6 Cal.
C
(pJ/K)
32 4
96 20
-
4
0.3
Fritz
C
(pJ/K)
46 ± 9
131±17
56
10
47.1+ 0.3
60
2
45.3 ± 0.2
Table 5.2: Heat capacity data taken with relaxation method. Note that the empty
calorimeter results ("Empty Cal. C") are an upper limit due to the low LR-400
bandwidth, and are for 68 mg of N grease. The K3 Fe(CN)6 calorimeter results
("K 3 Fe(CN) 6 Cal. C") are for 6.44 mg K3 Fe(CN) 6 and 13.7 mg N grease. The
"Fritz C" column is the heat capacity of 6.44 mg of K3 Fe(CN) 6 from the Fritz paper.
See text for explanation of the "errors" in the Fritz data.
the few percent level. These problems with temperature control were overcome in
the Sr 3 CuPtO.5 1ro. 5 0 6 experiment described in the next chapter, through the use of
superior instrumentation and due to the fact that thermal time constants were 3s or
less.
Now I compare the AC and Fritz data at zero field. The agreement for temperatures near TN is quite good, but at higher temperatures the AC data are significantly
higher than the Fritz data. Furthermore, this enhancement above TN is confirmed
by the relaxation data. Given the results of the transfer function analysis and the
agreement between AC and relaxation data, it seems reasonable to conclude that
D = 1 over the range of temperatures measured. If that is the case, then the physics
of the K3 Fe(CN)6 sample measured here must be somehow different from that of the
samples reported on by Fritz.
The data reported by Fritz were taken on single crystals of K 3 Fe(CN)6 grown from
aqueous solutions of analytical reagent grade K3 Fe(CN) 6 . As noted in Section 5.1,
the material used here was > 99% pure single crystals, crushed with mortar and
pestle. It is well-known that air and light (particularly UV) will slowly convert
K3 Fe(CN) 6 to K4 Fe(CN) 6 . K 4 Fe(CN) 6 contains Fe2 +, which has spin two in many
circumstances [1] [2]. The samples used here were protected from light by covering
in blackcloth or storing in dark cabinets, and were exposed directly to air for only
one or two days. Little conversion of K3 Fe(CN) 6 to K4 Fe(CN)6 is expected under
these circumstances [56]. However, the conversion may have been enhanced since the
K3 Fe(CN) 6 was in powder form after it was crushed and mixed with N grease.
One hypothesis regarding the difference between my sample and that of Fritz
is based on the idea that the heat capacity above TN is determined by short-range
ordering within the chains, and that at TN by long-range ordering. Then one would
say that the short-range ordering is somehow enhanced in my sample relative to
that of Fritz, whereas the long-range ordering is affected very little. Suppose there
are impurities within the chains, for example sites with Fe2+rather than Fe3+. The
85
relevant parameters here are the impurity concentration x and the correlation length
(. For an Ising chain at low temperatures [2],
!e/kBT
2~
(5.2)
given J ~ 0.36K for K 3Fe(CN) 6 , this leads to ( ~ 2 at 0.3K, and ( ~ 8 at 0.13K.
Hence it seems that impurity concentrations x of 10 - 50% would be necessary to
produce dramatic changes in the short-range order. It is difficult to see how these
kinds of impurity concentrations could be present in my sample. Nor is it clear how
such impurities would enhance the heat capacity due to short-range order. One might
argue that if Fe 2 is indeed in a spin two configuration, it would contribute extra spin
entropy to the system, which might be removed through short-range ordering. Another argument might be to point out [1] that superexchange, which is the source
of nearest neighbor coupling in K3 Fe(CN) 6 , is very sensitive to the relative positions
of surrounding atoms and to the non-magnetic species participating in the superexchange. Hence, if some of the CN groups were replaced by impurities or water, or
if the crystal structure were somehow modified (say by grinding), the superexchange
coupling might be made less anisotropic. This would lead to a more Heisenberg-like
behavior, which is known to have an enhanced short-range order contribution to heat
capacity relative to Ising behavior. In any case, studies of impurities in TMMC [57]
show that even very small impurity concentrations, with 1/x ~3, can produce
downward shifts in TN of 3%, and impurity concentrations such that 1/x ~ lead to
pronounced broadening in the peak at TN in addition to the shift. Hence it would be
surprising if impurities in the chains enhanced the short-range order without affecting
the long-range order in some way.
In the end, it seems difficult to produce a simple physical model to explain the
differences between the AC and Fritz data at zero field. A definite disadvantage of
the use of K3 Fe(CN) 6 as a standard for heat capacity measurement is the complexity
and sensitivity of the superexchange interaction that dominates its low-temperature
behavior. On the other hand, it is clear that this apparatus detected the transition
to long-range order in K3 Fe(CN) 6 , that the AC and relaxation data agree reasonably
well to within (admittedly large) error bars, and that the AC data correlates well with
published data, particularly around TN. It is also apparent from the transfer functions and heat capacity data that the thermal properties of the calorimeter are sufficient for accurate AC and relaxation method measurements, and that field-dependent
measurements are feasible. This measurement of heat capacity of K3 Fe(CN) 6 gives
confidence that the apparatus will be able to detect any interesting features in the
Sr 3 CuPt. 5 Ir 0 .5 O6
spectrum, and suggests changes (such as improved temperature
control) that will improve the quality of the relaxation data.
86
Chapter 6
Sr 3 CuPtO.51ro. 5 0 6 Experiment
Given the favorable results of the K3 Fe(CN) 6 experiment, it was appropriate to move
on to measurement of Sr 3 CuPtO.5 IrO. 5 0 6 heat capacity. There were three goals for
the Sr 3 CuPtO.5 1r o .5 0 6 experiment: first, in light of the susceptibility measurements
of Beauchamp, to look for a transition to long-range order above 1 K ; second, to
measure zero field specific heat for T < 1K if there was no clear evidence for a
transition to long-range order; and third, to measure field-dependence of specific heat
to 70 kG. The first run of the Sr 3 CuPtO.5 Iro. 5 0 6 experiment had serious technical
problems that made it impossible to achieve these goals. Due to the NbTi lead wires,
three of the four thermometers lost at least one wire during cooldown, which meant
that one four-wire thermometer was available on the empty calorimeter and two
three-wire thermometers were available on the Sr 3 CuPtO.5 1r o .5 0 6 calorimeter. This
made it impossible to measure temperature and ITac accurately. Also, there was
sufficient K3 Fe(CN) 6 residue (less than a few pg!) in the "empty" calorimeter to lead
to a K3 Fe(CN) 6 signature at 130 mK. Due to these difficulties, all the data reported
below were taken during the second run of the Sr 3 CuPtO.5 1r o .5 0 6 experiment, in which
these technical problems were solved as described in Section 6.1.
6.1
Calorimeter Preparation
As for the K 3Fe(CN) 6 experiment, calorimeter preparation began with two complete calorimeter plates. These were the same plates used for the K3 Fe(CN)6 experiment.
The Sr 3 CuPtO.5 Iro.5 O6 sample was mounted on the two plates formerly
used for the K3 Fe(CN) 6 experiment empty calorimeter, while the empty calorimeter
for this experiment used the two plates formerly used for the K 3 Fe(CN)6 sample. The
K 3Fe(CN) 6 residue was mechanically removed from the plates using 20 pm (Number
600) silicon carbide grit. It was assumed that this change in surface texturing did
not change the thermal properties of the plates significantly. The NbTi wires of the
K3 Fe(CN)6 experiment were replaced with PtW wires, as discussed in Section 4.2.2.
105.6 mg of Sr 3 CuPtO.5 1r o .5 0 6 were used during this experiment. 73.1 mg of this
were applied using the method described in Section 5.1, and the rest using a new
technique. The new technique was developed because it was tedious to scrape all of
87
the dough out of the mortar and pestle. Sr 3 CuPtO.s
5 ro.0 O6 and N grease were measured
out on separate tares. The Sr 3 CuPtO.5 Iro.5
was ground by itself in the mortar and
pestle, then transferred onto the N grease tare and mixed with the grease. This
transfer was easy because the powder and mortar were dry at this point. The amount
of dough was measured, and the dough scraped from the tare to the calorimeter. Then
the final mass of tare plus dough residue was measured to determine how much of
the dough had actually been transferred to the calorimeter. A total of 238.5 mg N
grease was used in making the Sr 3 CuPt0
ro. 5 0
6
dough. 200.0 mg N grease was used
on the empty calorimeter.
6.2
AC Method Procedures
Two thermal transfer functions were measured for each calorimeter, one just above
100 mK and one at 400 mK, at zero field. The EPAF range chosen was 0.13 rad/s
to 25.13 rad/s. The method used to acquire and correct ITac| data was exactly as described in Section 5.2. As for the K 3Fe(CN) 6 experiment, the need to correct thermal
transfer functions for the LR-400 bridge transfer function was not recognized until
after the experiment was completed. Hence the bridge again distorted the transfer
functions at high frequency, and the frequencies chosen for heat capacity measurements were below plateau for both calorimeters. For the empty calorimeter, 2.51 rad/s
was chosen for heat capacity measurements (for all fields and temperatures). For the
Sr 3 CuPtO.5 IrO.5 O6 calorimeter, 1.26 rad/s was chosen (for all fields and temperatures).
Fortunately, these choices were not very far below the plateau (of the corrected transfer function), and a plateau region is visible in most of the corrected transfer functions.
Hence it is possible to correct the zero field heat capacity data for the poor frequency
choice (see Section 6.5). Nevertheless, due to the uncertainties involved in making
this correction, the heat capacity data taken with the AC method will only be used
as a check on the relaxation data, and only data taken with this latter method will
be analyzed to determine the specific heat of Sr 3 CuPtO.s5 ro.5 O6 .
In addition to zero field, AC heat capacity was measured for both calorimeters in
fields up to 5 kG.
6.3
Thermal Relaxation Method Procedures
As mentioned in Section 5.3, Kb and AT(t) alone are useful for calorimeter thermal
characterization. For the Sr 3 CuPtO.5 Iro.50
6
experiment an additional thermal charac-
terization experiment was possible. Since all calorimeter thermometers and heaters
except for the Sr 3 CuPtO.5 1ro.5 0 6 calorimeter top heater were functional for the en-
tire experimental run, it was possible to estimate the thermal conductance across
the dough by examining differences in power/temperature curve slopes for curves obtained with different heater/thermometer combinations. For the empty calorimeter,
power/temperature curves were measured at 128 mK and 400 mK using all combinations of heaters and thermometers. For two curves measured with the same ther88
mometer but different heaters, the difference in the slopes of these curves is related
to the dough thermal conductance via (see Appendix A)
Sbb
K7
(6.1)
mbt
mbb
where mbt is the slope of the bottom thermometer, top heater curve; and mbb the
slope of the bottom thermometer, bottom heater curve. (A similar expression holds
for the top thermometer curves, with mbt replaced by mtb and mbb replaced by mtt.)
Here it is assumed that the thermal link wires have the same thermal conductance.
For the Sr 3 CuPtO.5 1r o .5 0 6 calorimeter, power/temperature curves were measured at
three temperatures using the bottom thermometer and bottom heater in one case and
the top thermometer and bottom heater in the other (this was necessary because the
top heater was not functioning). In this case, K, is found from (see Appendix A)
Ks= mtb(( mtb - mbb ) 2
'mbb
+
2
mTn
- Mnbb_
mbb
(6.2)
is the slope of the top thermometer, bottom heater curve; and mbb the slope of
the bottom thermometer, bottom heater curve. Again, the thermal link wires are
assumed to have the same thermal conductance. The values of K, obtained using,
these equations will be discussed in Sections 6.4 and 6.5 below.
mtb
Since the Sr 3 CuPtO.5 1ro. 5 0 6 calorimeter top heater was not functioning on this run,
and since it was desired to take heat capacity data using heater and thermometer on
opposite sides of the dough to detect possible problems with dough thermal resistance,
the top thermometer and bottom heater were used for heat capacity data acquisition
for the Sr 3 CuPtO.5 IrO. 5 0 6 calorimeter. A few points with bottom thermometer and
bottom heater were taken to check for differences (see Section 6.5). For consistency
with the Sr 3 CuPtO.5 1r o .5 0 6 calorimeter, the top thermometer and bottom heater were
also used for heat capacity data acquisition on the empty calorimeter. A few points
with other thermometer/heater combinations were also taken on this calorimeter to
check for differences (see Section 6.4).
The method used for mixing chamber temperature control for this experiment
differed from, and was superior to, that used for the K3 Fe(CN) 6 experiment. The
LR-400 bridge was used to measure mixing chamber chip resistance. The desired
mixing chamber chip resistance was set directly on the LR-400 bridge, and the 1OAR
output of the bridge fed to the PARC Model 113 pre-amp. The pre-amp output was
fed into a home made temperature controller, the output of which drove the mixing
chamber heater. This setup differs from that of the K 3Fe(CN) 6 experiment in that
the LR-400 provides an amplified voltage, proportional to the difference between the
chip resistance and the setpoint resistance, directly to the PARC pre-amp. In the
K 3Fe(CN) 6 experiment (Section 5.3), the homemade bridge and lockin provided a
voltage proportional to the chip resistance, which then had to be subtracted from
the setpoint and amplified by the PARC pre-amp. In the Sr 3 CuPtO.5 1ro. 5 0 6 experi-
ment, once temperature control was established at a new temperature (which typically
required 10 minutes for a 100 mK temperature step), the mixing chamber chip resis89
tance fluctuated by less than +2 ohms at 6kQ and less than +1 ohm at 3kQ. This
corresponds to temperature fluctuations of 0.1mK at 130mK and 0.5mK at 400mK.
The power PO to be applied to the heater for AzT(t) relaxation data was determined
by the condition that the temperature change AT produced by PO be at least 10
times as large as the noise in the T measurement. With PO determined, Kb was
then determined using the K 3 Fe(CN) 6 experiment procedure that allowed for drift
correction; i.e. between every two or three points, the temperature at zero power was
checked, rather than ramping the power up to P and then back down. The drift was
corrected for in the analysis or, if it was too severe (total drift 10 Q or more during
the measurement was considered severe), the measurement was redone with better
settings on the PID controller. A typical power/temperature curve is shown below.
3.5 3.0 2.5 c 2.0 S1.5
1.0 Data
0.5 -
-Fit
0
0.191
0.193
0.195
Temperature [K]
0.197
0.199
Figure 6-1: Typical power/temperature curve for the Sr 3 CuPtO.5 ro.0 O 6 experiment,
with drift correction.
As mentioned in Section 5.3, the LR-400 bridge is not adequate for A T(t) measurement for time constants of a few seconds or less. In this experiment, time constants
of both the sample and empty calorimeters were less than 3 seconds over the entire
temperature range examined. To measure AT(t) for short time constants, the arrangement shown in Figure 6-2 was used. The Stanford Research Systems SR830
lock-in amplifier supplied a 3 Volt, 100 Hz AC voltage to a home made circuit. This
circuit supplied 100 Hz, 0.25pA AC current to the current leads of the thermometer
via a transformer, and sensed (with amplification provided by another transformer)
the resulting voltage on the thermometer voltage leads. This voltage was then measured with the SR830. 100 Hz was chosen because it maximized AT(t) signal-to-noise.
A 30 ms time constant was used for the SR830. This time constant was chosen as the
90
Mixing Chamber
Homemade
Bridge + SR8 30
Lock-In
RB
Home nade
Tempe rature
Contr )ller
LeCroy
Scope
O)ut
Current
Source
C
trl In
-
Figure
6-2:
Apparatus
PC
Linux Box
used
to
measure
Sr 3 CuPtO. 5 1ro. 5 0 6 experiment.
91
relaxation
heat
capacity
for
best tradeoff between minimum distortion of AT(t) (which argues for a short time
constant) and noise in AT(t) (which argues for a long time constant). However, even
this lockin time constant led to a fairly noisy AT(t). This problem was solved by
taking many (typically 10) AT(t) measurements and averaging them with a LeCroy
9410 digital oscilloscope. A typical AT(t) before and after signal averaging is shown
in Figure 6-3. The resulting averaged AT(t) was transferred to the PC for further
analysis.
I
*
0.260
0.258
0.256
5 0.254
E
9 0.252
A.
0.250
0.248
0
6
12
18
Time [s]
Figure 6-3: Raw AT(t) (bottom curve) and AT(t) after averaging ten decays (top
curve). Top curve is shifted up by 2 mK for comparison.
To check the homemade circuit for self-heating, the current supplied by the homemade circuit was reduced until the SR830 reading was found to be independent of
current. The 0.25 pA used was in this current-independent regime. Also, the LR400 was connected to a calorimeter thermometer, and no change was recorded on it
when the homemade circuit was connected to and disconnected from the calorimeter
thermometer on the opposite plate. The time response of this SR830 setup was also
checked. A simple circuit that allowed for manual switching between two different
resistances was constructed, and the response of the SR830 setup and the hp34401a to
switching was measured. The hp34401a found a switching time of less than 10 ms. For
the SR830 setup, the response was exponential with a time constant of 51 t 3 ms.
The measured AT(t) must be corrected for the finite response time of the SR830
setup. This matter will be discussed in Sections 6.4 and 6.5 below.
Note that the thermometer resistance is not measured directly in this measurement
scheme. However, measurements show a linear relationship between lockin voltage
and thermometer resistance. To convert lockin voltage to thermometer resistance,
92
the resistances at zero heater power and at Po, measured for the Kb determination,
are identified with the corresponding lockin voltages in the AT(t) data. Voltages
between these two are converted to resistance by linear interpolation. Then R(T),
known from the 3 He MCT calibration, is used to obtain AT(t) from AR(t).
6.4
6.4.1
Empty Calorimeter Results
AC Method Results
10-2
,o
* Measured
2 Wire Model, Exc t:
---- Schwall Model, Ex Vc t--- Published Material s Data(Zr1 V r
, S= P111 K
10-3
3
!:
*
*
/
/
/
/
2
10-4
0.1
0.2
0.5
1
2
5
10
20
50
100
co [rad/s]
Figure 6-4: Empty calorimeter 128 mK thermal transfer functions.
As usual, thermal transfer functions were measured on the empty calorimeter in
order to determine a proper EPAF for heat capacity measurements and for thermal
characterization of the calorimeter. 128 mK and 400 mK empty calorimeter thermal
transfer functions, corrected for the effects of the LR-400, are shown in Figures 6-4 and
6-5. The low-frequency knees are visible on each transfer function, and the plateau
is visible in the 400mK function. Since the scatter in the data is again large relative
to the assumed error, the transfer functions were analyzed as described in Section
5.4 to obtain estimates of K,. Both transfer functions were fit to the exact twowire model to obtain the calorimeter thermal conductance, and the resulting thermal
parameters used to obtain the SSGI transfer function. The model parameters were
determined as for the K 3Fe(CN) 6 experiment, with the exception of the calorimeter
heat capacity. This was taken to be the value measured with the relaxation method.
Due to observation of a T2 effect in the empty calorimeter relaxation data at low
93
0.1
0.05
0.02
S0.01
0.005
-2
0.001
0.1
0.2
0.5
1
Wire Model, Exact
-- Schwall Model, Exact
Published Materials Data
SSGI, K=100 uW/K
"/SSIK=10w/
0.002
Measured
2
5
10
20
50
100
w [rad/s]
Figure 6-5: Empty calorimeter 400 mK thermal transfer functions.
temperatures, these transfer functions were also fit to a Schwall-type model, discussed
in Appendix D. For this Schwall transfer function, there were two fit parameters: heat
capacity of lump one C1 , and the thermal conductance of lump one K 1 (this is not
taken as the sample conductance K, for reasons discussed in Section 6.4.2). The heat
capacity of the second lump C 2 was taken to be the value Ctrt measured with the
relaxation method, minus C 1 . The conductance of the second lump was taken from
the power/temperature curve results. As Figures 6-4 and 6-5 show, the agreement
between the models and the measured functions is good. The measured function
agrees better with the Schwall function at low temperatures, as expected from the
relaxation results, but the Schwall and two-wire models both describe the data well
at 400mK, also expected from the relaxation results (the T 2 effect vanishes above
350mK). The calorimeter sample conductance K, as determined by the two-wire
transfer function is shown in Table 6.1 (the fitted C 1 , K 1 from the Schwall transfer
function will be shown in Table 6.5). The K, value determined by the transfer function
fit is clearly unreasonable at 128mK. This is because the transfer function does not
extend to sufficiently high frequency to show the high-frequency knee. Without this
knee, it is only possible to put a lower limit on K. The empty calorimeter thermal
conductance K, was also measured by comparing power/temperature curves taken
with bottom thermometer, bottom heater on the one hand, and bottom thermometer,
top heater on the other. This was done at 127 mK and 394 mK. The resulting
power/temperature curve slopes, and the thermal conductance K, derived from them
using equation (6.1), are shown in Table 6.1. Also shown is a lower limit on K,
obtained by shifting the slopes appropriately by one standard deviation. Finally, a
94
Temp
(mK)
128
400
2WTF
Fit
(ptW/K)
> 1026
93
40
TF
Low. Lim.
(pW/K)
20
50
PT
Slope
(pW/K)
28
470
PT
Low. Lim.
(pW/K)
26
180
Kb/K
Range
0.10 - 0.13
0.03-0.08
Table 6.1: Estimates of empty calorimeter K,. K, was determined from a fit of
the transfer function to the two-wire model (2WTF Fit), and from the slopes of the
power/temperature curves and Equation (6.1) (PT Slope). Lower limits on K, were
determined by comparison of the data and two-wire transfer functions with various K,
(TF Low. Lim.), and by varying the power/temperature curve slopes appropriately
by one standard deviation (PT Low. Lim.).
range for Kb/K, is found from the largest and smallest possible K, shown in the table
at the given temperature. It should be noted that power/temperature curves were
also taken using top thermometer combinations, and the resulting slopes were the
same to within the error bars. The top thermometer data would then suggest that
the K, are much larger than any of those reported here; hence even the largest values
reported in the table must be considered lower limits.
In addition, transfer functions based on the two-wire model and published data on
the calorimeter materials were computed and compared with measured functions for
the empty calorimeter. The model parameters were determined as for the computed
empty calorimeter functions in the K3 Fe(CN) 6 experiment. The dough thickness was
assumed to be 10 t 5 mil. Figures 6-4 and 6-5 compare the computed with the
measured functions. Again, the agreement is relatively good. Since the computed
Kb is higher than measured, it is no surprise to see the computed function fall off
faster with decreasing frequency. The computed plateau is also at higher wITct than
the measured, due to the fact that the measured heat capacity is higher than the
computed. This also is not too surprising, since the materials used for the calorimeter
may have impurity heat capacity contributions not seen in the published results.
AC heat capacity data were taken for 163 mK < T < 430 mK at zero field.
In addition to the usual corrections for the lockin RMS reading and LR-400 transfer
function, these data were corrected for the fact that the chosen EPAF was below
the plateau of the corrected transfer function. To obtain the off-plateau correction factor, the ratios JTact(Wp)/ITac(Wmeas = 2.51 rad/s), for wp = 5.03 rad/s and
wp = 25.13 rad/s, were computed. These wp were chosen because they describe the
boundary within which the actual plateau would be expected to fall. Then the offplateau correction factor A was taken to be the mean of these two values, and was
assigned an error equal to half of the difference between the two values. Using this
method, A = 1.5 ± 0.1 at 128 mK, and A = 1.8 t 0.2 at 400 mK. To account for
the temperature dependence of A, a simple linear interpolation was done for temperatures between 128 mK and 400 mK. The resulting C/T is plotted in Figure 6-6,
along with the measured relaxation heat capacity. The agreement between the AC
95
and relaxation results is quite good. This suggests that D = 1 on plateau, consistent
with the conclusions of the transfer function analysis. Note that the error bars in
the AC data are dominated by the uncertainty in my determination of A. AC heat
--
2.00
.75 .50
.25 x
C)
.00
o OkG oc
* OkG relax
0. 75 k
0.50
-
0
-I
0.2
0.4
0.6
Temperature [K]
0.8
1.0
Figure 6-6: Empty calorimeter AC heat capacity with off-plateau correction, and
relaxation heat capacity.
capacity data were also taken in several fields up to 5 kG. Since thermal transfer
functions were not available for these fields, it was not possible to correct these data
for any off-plateau effect. Hence these data are plotted (along with the uncorrected
zero field data) without any off-plateau correction in Figures 6-7 and 6-8, and are
useful mainly for qualitative analysis.
6.4.2
Relaxation Method Results
Recall that heat capacity is obtained in the relaxation method (provided that the thermal properties of the calorimeter are not too bad) by measurement of Kb and AT(t),
and fitting the latter to one or a sum of two exponentials. In the Sr 3CuPtO.5 Iro.0 O6 ex-
periment, the finite response time of the SR830 setup must be taken into account in
order to fit AT(t) properly. As mentioned above, the SR830 setup had a step response
given by
t< 0
S 0
1 - exp(-t/Tr)
t > 0
(6.3)
where Tr, = 51 ± 3ms. Hence AT(t) is the convolution of the thermal response of
the calorimeter and this exponential response. In Appendix B, the expected output
96
* OkG relax
* 1kG relax
o OkG AC
o 1kG AC
30
U
2.5
-
X2.0
-
_
0
-,j
11~
0
0
0
0
0
1.5
0.5
EPI
0.35
0.20
Temperature [K]
0.05
.I
0.50
Figure 6-7: Empty calorimeter AC (no off-plateau correction) and relaxation heat
capacity, 0 kG and 1 kG.
3kG relax
5kG relax
3kG AC
v 5kG AC
-
3.0
2.5
-A
x
A
-
2.0
A
V
1.5
A
1.0
U0.5
0.05
0.35
0.20
0.50
Temperature [K]
Figure 6-8: Empty calorimeter AC (no off-plateau correction) and relaxation heat
capacity, 3 kG and 5 kG.
97
signal AT(t) given s(t) and the actual thermal response ATi(t) is computed for A Ti(t)
given by one and two exponentials. The measured AT(t) were fit to the appropriate
expression for r, = 48, 51, 54 ms. From these fits, the exponential coefficients A, B
and time constants TA, TB of the thermal response
AT(t) = Ae-'I'A + Be-''B
(6.4)
were obtained for each r.
For the power/temperature curves, heater currents again could be measured to
better than ± 0.1 pA, and the heater resistance was known to better than ± 2 Q,
yielding errors in power of typically t0.4 nW at the lowest temperatures and ±10 nW
at the highest temperatures (near 1 K). With these errors, x2 s less than or near one
were obtained for the linear fits of the power/temperature curves (less than because
the current measurement is probably better than the indicated error). A typical
power/temperature curve was shown in Figure 6-1 above. The Kb data in zero field
are plotted as a function of temperature along with the expected Kb in Figure 6-10.
The expected curve was computed based on Suomi's data [54] for commercial copper
wire. In Figure 6-11, field dependence of Kb is shown. It is expected that Kb will
be dominated by the electron channel in the copper link wire at these temperatures.
Given that the wire used here was common commercial wire, prepared without any
special attention to purity, it is likely that there are magnetic impurities in the wire
that could lead to significant magnetoresistance and a reduction in Kb at high fields.
Relaxation heat capacity was measured in zero field for 137 mK < T < 846 mK,
and in fields up to 70 kG for 130 mK < T < 450 mK. Above 350 mK, AT(t) fit best
to a single exponential. Below 350 mK, all AT(t) curves required two exponentials
with different time constants for a good fit (Figure 6-9). For the data above 350 mK,
heat capacity was computed as C = KbT. For the data below 350 mK, the Schwall
formula
AT A + BTB
B
Cot = Kb A
(6.5)
was used. Use of this formula is discussed further below. With typical scatter in
AT(t) of ± 0.03 mK, x2 s close to one were obtained for the fits. The mean heat
capacity was taken from Kb and the T, = 51 ms fit results. The error in the heat
capacity was found by propagating the errors in the Kb and r, = 51 ms AT(t) fit
through the heat capacity formulae, and adding to this result half the difference
between the heat capacities obtained for T, = 48 ms and Tr, = 54 ms. As the field was
increased, especially above 10 kG, the larger time constant became still larger at low
temperature, while the smaller did not change or even fell slightly.
Relaxation heat capacity was measured at 127 mK and 394 mK at zero field with
all thermometer/heater combinations. These data are shown in Table 6.2. The differences in the heat capacity results due to the use of different thermometer/heater combinations are clearly larger than the errors due to the fits. This is thought to be due
to the asymmetry of the calorimeter design, in particular that the top plate is in contact with longer vespel pegs than is the bottom plate. Hence it is important that the
same thermometer/heater combination be chosen for both the Sr 3 CuPtO.5 Iro.50
98
6
and
129.0
128.5
~ Data
Two-exponential fit
--- One-exponential fit
o 128.0
-
127.5
E
g 127.0 -
\-
126.5
126.0
0
3
9
6
Time [s]
Figure 6-9: Empty calorimeter low temperature AT(t), showing best fit to sum of
two exponentials.
5
-
e
=3
-D
0
2
E
* Measured
Published Data
-
.*~
0
I-
0.1
0.3
0.5
Temperature [K]
0.7
_
0.9
Figure 6-10: Empty calorimeter zero field thermal conductance to bath Kb, measured
and predicted from published data on copper.
99
I
I
I
2.0
* OkG
o 5kG
* 10kG
o 30kG
A 50kG
70kG
1.5
e
x
.
00
.
-
Cc1.0
.
0
UA
00
0
AA
0*
~0.5
A
A
AA
AA
0
0.1
0.2
0.3
Temperature [K]
0.4
0.5
Figure 6-11: Field dependence of Kb.
Temp
(mK)
127 ± 1
392 + 1
Ctt
Ctb
Cbb
Cbt
(pJ/K)
1.39 ± 0.03
4.3 ± 0.3
(pJ/K)
1.60 ± 0.05
3.6 ± 0.06
(pJ/K)
2.03 ± 0.06
4.6 ± 0.5
(pJ/K)
2.16 ± 0.07
3.91 ± 0.07
Table 6.2: Relaxation heat capacity from various thermometer/heater combinations
on the empty calorimeter. The notation Cth refers to heat capacity measured with
thermometer t, heater h.
100
empty calorimeters, in order to obtain the appropriate amount to subtract from the
Sr 3 CuPtO.5 1r o .5 0 6 calorimeter result to obtain Sr 3 CuPtO.5 1ro. 5 0 6 specific heat.
It is somewhat surprising that the empty calorimeter heat capacity shows field
dependence at low fields. The materials chosen for the calorimeter were chosen in
part because they were thought to be non-magnetic. There are two possible reasons
for this magnetic response. First, some pure component in the calorimeter, thought
to be non-magnetic, actually is magnetic. Second, there are magnetic impurities in
some component(s) of the calorimeter. Given the size of the magnetic response (at
the lowest temperatures, C/T doubles between OkG and 1kG), it seems unlikely that
the PtW wires, chip resistors, or copper thermal link wire could be responsible for
the effect (see Figure 4-4). This leaves the quartz plates, N grease, or copper fingers.
Since N grease is composed of petroleum-based hydrocarbons [58], its specific heat is
unlikely to have any dramatic field dependence. Pure quartz is also non-magnetic.
As mentioned in Section 4.2.2, the copper fingers were made from commercial copper
shim stock, and so may contain large amounts of magnetic impurities such as iron.
It seems most likely that the copper fingers are responsible for this low-field heat
capacity response.
I return now to the question of whether it is appropriate to use the Schwall formula
to compute heat capacity for two-exponential fits with this calorimeter. Since two
exponentials fit the data best (both one and three exponentials give worse fits), it is
reasonable to model the calorimeter with two lumps, having heat capacities C1 and
C2, and connected by some thermal conductance K 1 . The only remaining question for
the model is then how to connect these lumps to thermal ground. It might seem most
appropriate, given the geometry of the calorimeter, to connect each to ground with a
thermal link Kb/2. Given this model, and measured values of Kb and the AT(t) fit
parameters A, B, TA, and TB, it is possible to predict C1, C2, and K 1 (see Appendix
C). Applying this model to the data gave unreasonable results. In particular, two
solutions for C1, C2, and K1 were obtained for each temperature examined. One
solution predicted K 1 < 0, and the other predicted C1 + C2 that did not extrapolate
smoothly to the higher temperature, one-exponential data (e.g. this model predicts
that C +C2= 5.5pJ/K at 322 mK, whereas at 351 mK, the lowest one-exponential fit
temperature, C + C2 = 3.18pJ/K). On the other hand, the standard Schwall model,
with only one of the lumps connected to ground, is found to give more reasonable
results: the K 1 are positive, and C1 + C2 is consistent with the high temperature
data. Furthermore, the results obtained with the standard Schwall model agree with
the corrected AC heat capacity(Figure 6-6).
Another argument against use of the two-link Schwall model can be made. Presumably, in the two-link model the two lumps most likely represent the two plates,
with the N grease in between providing K 1 = K,. However, in light of the other thermal characterization data, the source of this two-exponential behavior could not be
poor thermal conductance across the dough. The power/temperature curves taken
with different heater/thermometer combinations suggest that the thermal conductance K, between the given heater and thermometer is reasonably large. However a
small, weak thermal link to some part of the calorimeter that is not between them
would not be detected in these measurements, and such a link could be the source of
101
Field
Temp
(kG)
0
0
0
0
0
70
70
(mK)
128
128
221
322
400
129
200
C1
Cu Fing.
(pJ/K)
0.1
0.1
0.2
0.3
0.3
25.2
10.6
C1 , K 1
Vespel
(pJ/K),(pW/K)
0.2,0.01
0.2,0.01
0.5,0.04
0.8,0.07
0.8,0.07
-
C1
C2
K1
(pJ/K)
0.4 ± 0.lx
0.56
0.64
0.78
1.3 ± 1.1x
13.66
4.35
(pJ/K)
1.2 t 0.lx
0.99
1.56
2.17
2.3 t 1.1x
0.66
0.97
(pW/K)
1.0 ± 0.6x
0.61
1.15
1.13
17 ± 19x
1.40
0.75
Table 6.3: Parameters for Schwall model of empty calorimeter. The values with an
"x" next to them are from fits of the measured AC transfer function to the Schwall
model. The Cu Fing. column shows estimated heat capacity for two sets of copper
fingers. The Vespel column shows estimated heat capacity and conductance for the
Vespel pegs.
this two-exponential behavior.
Hence it seems appropriate to use the standard one-link Schwall model to compute
heat capacity of the empty calorimeter for the case where AT(t) fits two exponentials.
It is desirable to go further and try to connect such a model with the actual calorimeter
geometry. Since K 1 is not between the calorimeter plates, lump 2 would be associated
with the calorimeter plates and dough, and lump 1 with the weakly connected part of
the calorimeter. This weakly connected part could be all or part of the copper fingers,
the vespel pegs, or the PtW lead wires, since these parts of the calorimeter are not
between the heaters and thermometers. At 130 mK, the PtW wires have an estimated
total heat capacity of 7 nJ/K, orders of magnitude below the measured heat capacity
(1.55 paJ/K) at this temperature and below the resolution of the experiment. The
vespel peg heat capacity at 130 mK is difficult to estimate, since no measurement of
vespel's heat capacity could be found in the literature; assuming it behaves like N
grease, the 6 pegs (massing less than 30 mg each) would have a total heat capacity of
0.3 pJ/K. Finally, the two sets of copper fingers have a total heat capacity (estimated
from the literature) of 0.12 pJ/K at 130 mK. These heat capacity estimates suggest
that the vespel pegs or copper fingers are the most likely candidates for lump 1. The
values of C1, C2, and K 1 derived from the AT(t) fit parameters and Kb for the onelink Schwall model are shown in Table 6.3 for zero field and 70 kG data. Also shown
(values are marked with an "x") are C 1 , C2, and K 1 as determined from the Schwall
transfer function introduced in Section 6.4.1.
Neither the computed copper finger nor the computed vespel numbers agree very
well with the numbers derived from the data. Given that the Schwall model is not very
detailed and only gives reliable results for Ctot, this is not too surprising. However,
for the zero field data, it seems more likely that the vespel pegs are responsible for the
two-exponential behavior. The heat capacity estimates are closer, and it is difficult
102
to imagine how K 1 could be so low for the copper fingers. For the 70 kG data, the
copper fingers are most likely to be responsible for the two-exponential behavior, since
copper is known to have a large nuclear contribution to its heat capacity at these
temperatures and fields. As shown in the table, the measured C1's here compare
reasonably well with the expected nuclear heat capacity of copper.
The fit parameters for the Schwall AC transfer function compare reasonably well
with those found from the relaxation data, particularly at 128mK. At 400mK, the
Schwall AC fit parameters have large errors, indicating that that model does not
work as well at high temperatures, which is consistent with the fact that no T 2 effect
was observed at high temperatures. This vanishing of T2 effect at high temperatures
is consistent with the Schwall relaxation model assuming that C2 > C1 at high
temperatures. This is reasonable, as one would expect the heat capacity of the quartz
plates to rise more quickly than that of Vespel as the temperature rises.
Finally, C/T for this empty calorimeter is at least a factor of three higher than
C/T of the empty calorimeter used for the K 3Fe(CN) 6 experiment (recall that C/T
measured for the latter experiment is an upper limit). This can be attributed to the
fact that 200 mg of N grease were used for this calorimeter, while only 68 mg were
used for the other.
6.5
6.5.1
Sr 3 CuPt 0 .Ir 0 .5 0 6 Calorimeter Results
AC Method Results
103
9
8
7
Measured
-2 Wire Model, Exact
SSGI, K,=20 AW/K
.
6
5
4
.. . .. . . .
3
104 1
0.1
0.2
0.5
1
2
5
10
20
sa
100
w [rad/s]
Figure 6-12: Sr 3 CuPto.5 1r o .5 0 6 calorimeter 136 mK thermal transfer functions.
103
0.01
0.009
0.008
0.007
0.006..
.. ..
0.005
0.004
0.003
2 Wire Model, Exact
-.-.-.SSGI, K,=37 AW/K
0.002
0.001
0.1
0.2
0.5
1
2
5
10
20
50
100
w [rod/s]
Figure 6-13: Sr 3 CuPtO.5 IrO.5 0 6 calorimeter 400 mK thermal transfer functions.
For the Sr 3 CuPtO.5 1ro.5 0 6 calorimeter, thermal transfer functions were measured
at 136 mK and 400 mK. These are shown in Figures 6-12 and 6-13, corrected for
the effects of the LR-400. In the frequency range examined, the low-frequency knee,
plateau, and the beginnings of the high-frequency knee are visible in each transfer
function. Again, the functions were compared to the exact two-wire model in the
manner described in Section 5.5 to obtain calorimeter thermal conductance, and the
resulting thermal parameters used to compute the SSGI transfer function. A fit to
the Schwall transfer function was attempted, but the fit parameters for C1 and K 1
were negative. Thus the Schwall model was excluded as a possible model for the
Sr 3 CuPtO.5 IrO. 5 O6 calorimeter in this temperature and field regime, which is consis-
tent with the one-exponential behavior found with the relaxation method (see 6.5.2).
Model parameters were determined as for the empty calorimeter, and all parameters
for the best-fit transfer functions are shown in Table 6.4. Calorimeter thermal conductance was also measured by comparing power/temperature curve slopes measured
with bottom thermometer and bottom heater on the one hand, and with top thermometer and bottom heater on the other. K, was then obtained using equation 6.2.
The calculated K, along with the lower limit on K, obtained by shifting the slopes
appropriately by one standard deviation, are shown in Table 6.4. Given the presence
of the plateau, and the low Kb/K, indicated in the table, D = 1 would be expected
for the AC heat capacity data.
Zero field AC heat capacity data were taken for 0.100 K < T < 2.1 K.
Some of thes data, including all of the data above 0.5 K, were taken during slow
cooldowns of the refrigerator from
-
2 K.
The lockin RMS reading and LR-400
transfer function corrections were applied. As for the empty calorimeter, the exci104
Temp
(mK)
136
TF
Low. Lim.
([LW/K)
PT
Slope
(pW/K)
PT
Low. Lim.
(pW/K)
Kb/K
Range
6
15
-
-
0.14-0.24
4
30
41
370
1400
25
130
193
0.1-0.15
0.03 - 0.08
0.38-0.52
0.01 - 0.09
2WTF
Fit
(pW/K)
19
145
306
400
430
-
37
-
-
Table 6.4: Sr 3 CuPtO.5 Iro.5 0 6 calorimeter K,. K, was determined from a fit of the
transfer function to the two-wire model (2WTF Fit), and from the slopes of the
power/temperature curves and Equation (6.1) (PT Slope). Lower limits on K, were
determined by comparison of the data and two-wire transfer functions with various K,
(TF Low. Lim.), and by varying the power/temperature curve slopes appropriately
by one standard deviation (PT Low. Lim.).
tation power frequencies chosen for heat capacity measurement were below plateau
for the Sr 3 CuPtO.5 1ro. 5 0 6 calorimeter. However, given that the plateaus are visible
in the measured transfer functions, it is again possible to correct for the poor choice
of frequency. For the 136 mK transfer function, the method described for the empty
calorimeter could not be used to determine the off-plateau correction factor A, due
to the scatter in the data at high frequencies. However, the model functions as well
as the measured function support the choice A = 1.0. For the 400 mK function,
the plateaus shown by the two model functions were used to estimate the range over
which the plateau could occur, giving A = 1.2 ± 0.1. Again, intermediate temperatures were corrected using linear interpolation. The resulting C/T is plotted in Figure
6-14, along with the measured relaxation heat capacity. The agreement between the
AC and relaxation results is quite good below 400 mK. This suggests that D = 1
in this temperature range, again consistent with the transfer function analysis. The
divergence between the AC and relaxation data at higher temperatures suggests that
the growth of A with temperature is stronger than linear.
AC heat capacity data were also taken in fields up to 5 kG, for temperatures
140 mK < T < 440 mK. As for the empty calorimeter, it was not possible to correct
these data for off-plateau effects. These data are plotted, along with uncorrected zero
field data and relaxation data, in Figures 6-15 and 6-16. Comparison of the AC data
with the low-field relaxation data (Figure 6-15) shows good qualitative agreement, and
that the AC data is slightly higher than the relaxation. This quantitative difference
is expected due to the off-plateau effect.
The zero field data above 1 K merits special attention, since this is the range in
which Beauchamp has seen a peak in AC susceptibility. Figure 6-17 shows this data,
taken on two cooldowns, with correction for the LR-400 transfer function applied.
Note that no correction for the off-plateau effect is applied, so Figure 6-17 shows
C/T/D, not C/T as is shown in Figure 6-14. This implies that the analysis of this
105
I
I
I
1.4
o OkG ac
* OkG relax
1.2
1.0
I 0.8 F
0
0
0
x
-
U-
0.6
|-
0.4 I-
ieTTTf
-
0-
0.2
0
0.4
0.8
Temperature [K]
1.2
1.6
Figure 6-14: Sr 3 CuPtO.5 Iro.0 O6 calorimeter AC heat capacity with off-plateau correc-
tion, and relaxation heat capacity data.
1.2
* OkG relax
* 1kG relax
OkG AC
0 1kG AC
0
1.0
0o
10
0.8
U~
x
0.6
r
00
o
0
0
0
o 0.4
-
OIQO
U
CbEm
0.2
F
0
0.1
0.3
0.5
Temperature [K]
0.7
0.9
Figure 6-15: Sr 3 CuPtO.5 Iro.5 O 6 calorimeter AC (no off-plateau correction) and relax-
ation heat capacity, 0 kG and 1 kG.
106
10 --
3kG relax
5kG relax
AC
v 5kG AC
9 -3kG
LO
XA
-2
A
A
A
8
V
-
c
6
6VA
V
A
A
V
V.AA
V V
0
A
A
y
3
0.05
0.35
0.20
0.50
Temperature [K]
Figure 6-16: Sr 3 CuPtO.5 1ro. 5 0 6 calorimeter AC (no off-plateau correction) and relax-
ation heat capacity, 3 kG and 5 kG.
data is mainly qualitative, which is sufficient for the purpose of detecting the presence
or absence of a feature. It is apparent that if there is a feature in heat capacity, it
is small. The region around 1.5 K appears most promising. To examine the excess
heat capacity in this region, the 21 November data points on the wings of the region
(0.7 K < T < 1.23 K and 1.78 K < T < 2.10 K) were first fit to an eighth
order polynomial. This polynomial was then subtracted from the data in the central
region, and the result is plotted in Figure 6-18. Error bars are just those from the C
data. The result suggests that there is a peak here. This will be discussed further in
the Section 7.1 below.
6.5.2
Relaxation Method Results
Relaxation heat capacity was measured in zero field for 137 mK < T < 846 mK,
and in fields up to 70 kG for 130 mK < T < 450 mK. AT(t) curves fit to a sum of
two exponential decays (plus the SR830 setup response) always yielded a single time
constant for H < 10 kG. Hence these curves were best fit to a single exponential.
Typical scatter in the curves was ±0.03 mK, leading to x 2 near one for the fits. Due
to the Sr 3 CuPtO.Or o.5 0 6 , the heat capacity C 2 of the calorimeter dominated that of
any second lump C 1 , so no two-exponential behavior was observed in this regime. To
check this against the model, the values for C1 and K 1 found at 130 mK in zero field
for the empty calorimeter were assumed for the Sr 3 CuPtO.5 IrO. 5 0 6 calorimeter, and
C 2 was taken to be KbT for the Sr 3 CuPtO.5 Iro.50 6 calorimeter at 140 mK. The model
107
1.50
-
-
1. 25
-
-3
-1. 00
x
0. 75
o
-
0.50
11/21 AC data
0 12/29 AC data
"
0 0
0.25 -00-
.
C,
0.6
I
1.4
1.8
Temperature [K]
1.0
2.2
2.6
Figure 6-17: Sr 3 CuPtO.5 Iro.0 O 6 calorimeter AC heat capacity data above 1 K.
* 11/21 AC data
o 12/29 AC data
2.0
1.5 -
1.0 -
IC:)
0.5 -
0
-0.5
1.2
1.3
1.4
1.5
Temperature [K]
1.6
1.7
1.8
Figure 6-18: Excess Sr 3 CuPtO.5 ro.5 06 calorimeter AC heat capacity data above 1 K
(see text).
108
Field
(kG)
70
70
70
Temp
(mK)
133
201
408
C1 Cu. Fingers
(pJ/K)
25.2
10.6
2.6
C1
(pJ/K)
16.74
8.26
1.248
C2
(pJ/K)
1.28
1.27
3.032
Table 6.5: Parameters for Schwall model of Sr 3 CuPtO.5 Iro.50
Temp
(mK)
144.9
148.3
305.9
302.8
Ctb
Cbb
(pJ/K)
(pJ/K)
11.0
-
K1
(pW/K)
1.60
1.097
0.424
6
calorimeter.
0.1
10.7 ± 0.2
-
-
14.9 i 0.2
15.5 ± 0.3
-
431.8
-
16.8 + 0.3
429.0
17.1 + 0.4
-
Table 6.6: Relaxation heat capacity from various thermometer/heater combinations
on the Sr 3 CuPtO.5 IrO. 5 0 6 calorimeter. The notation Cth refers to heat capacity measured with thermometer t, heater h.
then predicted that at least 97 % of the throw should be in a single time constant,
consistent with observation. However, above 10 kG, a long, second time constant
appeared. As for the empty calorimeter, this second time constant is thought to be
due to the nuclear heat capacity of the copper fingers. Values of C1, C2, and K1 for
the Sr 3 CuPtO.5 IrO. 5 0 6 calorimeter at 70 kG are shown in Table 6.5. It is reassuring
that the K 1 are comparable to those obtained for the empty calorimeter at 70 kG
(Table 6.3), and that the C2 are comparable but larger.
The error analysis for the power/temperature curves here is identical to that for
the empty calorimeter, with the same typical errors in power measurement and x 2
of one typical for the fits. The Kb data in zero field are plotted as a function of
temperature along with the expected Kb in Figure 6-19. The expected curve was
computed based on Suomi's [54] data for commercial copper wire. In Figure 6-20, the
field dependence of Kb is shown.
As a check, relaxation heat capacity was measured for a few points at zero field using the bottom thermometer and bottom heater. The results were consistent with the
usual top thermometer, bottom heater data to within the error bars of the respective
measurements, as shown in Table 6.6.
109
5
LOl
C-)
0
-
* Measured
Published Data
*
0
0.1
0.3
0.5
Temperature [K]
0.7
0.9
Figure 6-19: Sr 3 CuPtO.5 Iro.0 6calorimeter zero field thermal conductance to bath Kb,
measured and predicted from published data on copper.
I
I
I
'
10l
2.0
U)1. 5
*
o
m
o
OkG
5kG
10kG
30kG
.
A5OkG
S
*
05-
70kG
x
CU
C-)
00.
U
-3
A
080
_0
|
-
-
0
S
0.1
.
I
I
0.2
0.3
Temperature [K]
.
I
0.4
0.5
Figure 6-20: Field dependence of Kb, Sr 3 CuPtO.5 1r o .5 0 6 calorimeter.
110
6.6
Sr 3 CuPtO.5 1ro.5 0 6 Specific Heat Determination
Given the heat capacities of the Sr 3 CuPtO.5 Iro. 5 0 6 and empty calorimeters, and the
mass of Sr 3 CuPtO.5 1ro. 5 0 6 used, the specific heat a of Sr 3 CuPtO.5 1r o .5 0 6 was computed
as a function of temperature and field. For all temperatures and fields, only the
relaxation data was considered for determination of Sr 3 CuPtO.5 Iro. 5 0 6 specific heat.
For fields of 10 kG and below, the heat capacity of the Sr 3 CuPtO.5 1ro.5 0 6 sample was
computed by simple subtraction of the Sr 3 CuPtO.5 1ro. 5 0 6 and empty calorimeter heat
capacities, and the sum of the errors in the two calorimeter heat capacities taken
There was a slight (38.5 mg)
as the error in the Sr 3 CuPtO.5 1r o .5 0 6 heat capacity.
difference in the amount of N grease in the two calorimeters, but this was smaller
than the error bars and so was neglected. Above 10 kG, the low amplitude, long
time constant component of AT(t) occasionally produced large errors in the AT(t)
fit, rendering a simple subtraction meaningless in some cases. Hence these data were
examined by plotting the empty and Sr 3CuPtO.5 1r o .5 0 6 calorimeter data on the same
plot, Figure 6-21.
104
5
* Sample cal. 70kG
0 Empty cal. 70kG
M Sample cal. 50kG
o Empty cal. 50kG
:
Sample cal. 30kG
Empty cal. 30kG
.
A
2
0A
S, 10-5
e
_'1
-1
.--
:
-
Set Cu fingers,70kG
1 Set Cu fingers,50k!
1 Set Cu fingers,30kG
5
0
2
ci6
5
2
107
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8 0.9 1
Temperature [Kelvin]
Figure 6-21: High field heat capacity of Sr 3 CuPtO.5 1ro. 5 0 6 and empty calorimeters.
The main conclusion to be drawn from the high field data is that the heat capacities of the Sr 3 CuPtO.5 Iro. 5 0 6 and empty calorimeters are essentially the same at these
fields. The Sr 3 CuPtO.5 r o.50 6 is slightly higher, which makes sense as a small phonon
contribution is still expected from the Sr 3 CuPtO.5 1ro. 5 0 6 . These data support the
hypothesis that the two calorimeters, measured with the same heater/thermometer
combination, are essentially the same modulo the magnetic heat capacity contri-
bution of the Sr 3 CuPto.5 1r o.5 0 6 .
This hypothesis is essential in order to obtain
111
Sr 3 CuPtO.5 I r o .50
6
heat capacity by simple subtraction of the two calorimeter heat
capacities.
For given mixing chamber temperatures, small differences were seen between the
temperatures of the empty and Sr 3 CuPtO.5 Iro.50 6 calorimeters. Below 300 mK, these
differences were always less than 3 mK, increasing to 5 mK at 375 mK and to
as much as 50 mK at Sr 3 CuPtO.5 I r o .50 6 calorimeter temperature of 850 mK. The
Sr 3 CuPtO.5 IrO. 5 0 6 calorimeter, in the position closest to the mixing chamber, was al-
ways at the higher temperature. Since heat is applied at the mixing chamber for
temperature control, and the amount of heat needed increases with mixing chamber
temperature, a temperature gradient is expected across the calorimeter stack, with
higher temperatures closer to the mixing chamber. When there were differences between the two calorimeter temperatures, linear interpolation was used to determine
the proper empty calorimeter heat capacity at a given Sr 3 CuPtO.5 IrO.5 O6 calorimeter
temperature before subtracting the two results. Errors in measured heat capacities
were propagated through the interpolation formula to give the final error in the empty
calorimeter heat capacity.
112
Chapter 7
Conclusions and Future Work
7.1
Sr 3 CuPt 0. 5Ir 0 .5 0
6
Specific Heat Results
In this section, I will restrict myself to presentation and analysis of the data, and will
only refer to specific interpretations of the data where necessary. Figure 7-1 shows
the zero-field u/T of Sr 3 CuPt. 5 1ro. 5 0 6 . u/T is plotted rather than the specific heat a
because the former is equal to the temperature derivative of entropy, ds/dT. The zerofield u/T data were integrated using a simple trapezoidal rule integration, and gave
0.12 J/mol/K for the change in entropy over the temperature range shown. One mole
of Sr 3 CuPtO.5 1r o .5 0 6 has 1.5 spins-1 per formula weight, or 1.5 R In 2 = 8.6J/mol/K
entropy content at infinite temperature. Hence the u/T data shown here accounts
for only 1.4 % of the total spin entropy.
In Figure 7-3, a for Sr 3 CuPtO.5 1r o.5 0 6 is shown directly. The fact that a decreases
monotonically while u/T increases with decreasing temperature below 0.4 K indicates
the possibility that a ~T, where 0 < 6 < 1, below 0.4 K. The a/T data below
0.4 K were fit to the RQSC expression
a/T = AT5-1 ln T/To1
(7.1)
The fit is shown in Figure 7-1; 6 = 0.50 ± 0.07 is found, with X2 = 4.0. The data
above 0.4K seem to have more scatter than that below. It was found experimentally
that temperature control was more difficult at higher temperatures, and this is one
cause of the increased scatter. However, a power law fit to all of the data (above
0.4K as well as below) led to a/T oc T-', where a = 0.76 ± 0.04, with X2 = 15. In
Figure 7-2, I show the fit to u/T = OT 0 76 +B, where # = 0.100±0.005J/mol/K' 2 4 ,
and B = -0.02 ± 0.01.
Figure 7-4 shows the field dependence of a/T to 10 kG. The 10 kG a data were
fit to the power law a = ATT', with the result -y = 1.52 t 0.07, A = 0.22 ± 0.02, and
3 2
X2 = 2.8. Figure 7-5 shows the 10 kG a data as a function of T / , along with the
5 2
3
best fit to a = A T /2+B. A = 0.22± 0.01 J/mol/K / , B = -0.0003±0.001 J/mol/K
are found, and X2 = 3.0. In Figure 7-6 a(H) is shown at 130 mK. This does not fit
to a power law (C oc H-) (0 = 0.6 ± 0.1 was the best fit, with X2 = 64!), but does
fit a second-order polynomial fairly well. This is shown in Figure 7-6. With X2 = 7,
113
0.45
* Data
-Fit <0.4K data RQSC
0.30
0. 15
00
0
0.1
-
0.3
-
0.5
0.7
0.9
Temperature [K]
Figure 7-1: Zero field c-/T for Sr 3 CuPtO.5 1ro. 5 0 6 , fit below 0.4K.
0.45
I
I
I
* Data
-Power
Law Fit
0.30
00
E
0 15
0 --
0.1
0.3
0.5
0.7
0.9
Temperature [K]
Figure 7-2: Zero field a/T for Sr 3 CuPtO.5 1r o .5 0 6 , fit to all data.
114
0.09
0
0.06
H
E
i
0
.0
0-
0.03 -
V)I
I
0
-
0.3
0.1
0.7
0.5
Temperature [K]
0.9
Figure 7-3: Zero field specific heat for Sr 3 CuPtO.5 1r o.5 0 6 .
0.45
I
I
I
I
I
T,
OkG
1kG
3kG
5kG
7kG
10kG
*
0.30 F
0
b
0. 15
P-
I
I.
0
0
0
0
0.1
I
I
0.3
0.5
Temperature [K]
I
0.7
Figure 7-4: Low field -/T for Sr 3 CuPtO.5 IrO. 5 0 6 .
115
I
0.9
I
I
I
I
I
I
I
I
Relaxation Dato
0.07 -Fit
0.06
0.05
0
0.04
-E
0.03
0.02
0.01
0
I
I
I
I
0
0.05
0.10
0.15
I
0.20
T3/2 [K3/ 2]
I
I
I
0.25
0.30
0.35
Figure 7-5: a at 10kG for Sr 3 CuPtO.5 1ro. 5 0 6 , and fit to a = AT 3/2 + B.
the fit was
a(H) = 0.05t0.002J/mol/K-(0.0063t0.0006J/mol/K/kG)H+(21.2x10~ 5J/mol/K/kG 2 )H 2
(7.2)
7.2
Discussion I: Entropy, Field Dependence
I mentioned in the last section that the total spin entropy of Sr 3 CuPtO.5 IrO. 5 O 6 should
be 1.5 R In 2, or 8.6 J/mol/K. 0.12 J/mol/K, or 1.4%, is removed in the temperature
region studied. If it is assumed that u/T continues to follow the T6- 1 In TI fit to the
T < 0.4K data down to T = 0, the total amount of entropy removed from the system
below 1K would still be only 3.1% of 1.5Rln2.
Since my relaxation data extends only to about 1K, and since the AC data taken at
higher temperatures is only useful for qualitative analysis, it is difficult to determine
the amount of entropy removed above 1K on the basis of my data alone. However,
the AC data can be used to estimate an upper limit on the amount of spin entropy
removed between 1K and 2K. If I take the AC data shown there to be entirely due to
Sr 3 CuPtO.5 IrO.5 O 6 , entirely magnetic, and ignore any off-plateau effect, I find 7% of
R In 2 there. Also, as noted in Chapter 1, Ramirez has taken heat capacity data on
Sr 3 CuPtO.5 1r o.5 0 6 and on diamagnetic Sr 3 ZnPtO 6 at temperatures from 2.5K to 50K.
From this data, it should be possible to determine roughly how much of the magnetic
entropy of Sr 3 CuPtO.5 1ro. 5 0 6 is removed on that temperature range. Ramirez's data
116
0.06
Relaxation data
0.05 -
0.04
0
E 0.03
0.02
0.01 -
0
0
7.5
Field [kG]
5.0
2.5
10.0
12.5
15.0
Figure 7-6: Specific Heat of Sr 3 CuPtO.5 1ro. 5 0 6 as a function of field at 130 mK.
were taken on 0.2g Sr 3 CuPtO. 5 IrO. 5 0 6 and 0.2g Sr 3 ZnPtO6 samples [32]. Each sample
was mixed with the same mass of silver powder and compressed to improve sample
thermal conductance.
To estimate the magnetic entropy removed between 2.5K and 50K, note first that
the lattice contribution dominates Sr 3 CuPtO.5 Iro.5 0 6 and Sr 3 ZnPtO6 heat capacities
over most of the temperature range, and that the heat capacity of Sr 3 ZnPtO6 is
slightly higher than that of Sr 3 CuPtO.5 1r o .5 0 6 at the highest temperatures (see
Figure 2-9). The former observation indicates that the lattice heat capacity of
Sr 3 CuPtO.5 1ro.5 0 6 must be determined and subtracted from the total to determine
the magnetic heat capacity. The latter observation indicates that the lattice heat
capacity of Sr 3 CuPtO. 5 1ro. 5 0 6 is lower than that of Sr 3 ZnPtO6 (since Sr 3 ZnPtO6 is a
diamagnetic insulator, all of its heat capacity is due to the lattice). Hence a simple
subtraction of the integrals of -/T for the two materials, which would assume equal
lattice heat capacities, will underestimate the magnetic entropy removed. A better
estimate can be obtained using the method of Stout and Catalano [59]. Given that
the crystal structures of Sr 3 CuPtO.5 IrO. 5 0 6 and Sr 3 ZnPtO6 are very similar, the law
of corresponding states would imply that the two lattice contributions can be written
Czpt =
Clcp
=
f(T/lOz)
(7.3)
f(T/lOpi)
(7.4)
that is, the two heat capacities are identical modulo a scaling of the temperature
axis. At sufficiently high temperatures, the total heat capacity will be just Cat
117
for both materials. Therefore the appropriate temperature scaling factor can be
determined at high temperatures, and the Sr 3 ZnPtO6 data multiplied by this factor to determine Ct. If the law of corresponding states is applicable, this scaling
factor should not change with temperature (at high temperature). Assuming that
50K is sufficiently high temperature for Sr 3 CuPt0 .5 Iro.5O 6 and Sr 3 ZnPtO6 , it is found
that a scaling factor of 1.07 overestimates the 50K Sr 3 CuPt. 5 Iro*O 6 heat capacity,
while a scaling factor of 1.1 underestimates it. Applying these scaling factors to the
Sr 3 ZnPtO 6 data and integrating under the resulting c-/T curves, and subtracting the
results from the integral under the Sr 3 CuPt. 5 Iro.5O 6 curve, I obtain 5 J/mol/K as a
lower limit, and 8 J/mol/K as an upper limit for the magnetic entropy removed from
Sr 3 CuPtO.5 1ro. 5 0 6 between 2.5K and 50K. These limits correspond to 58% and 93%
of the total magnetic entropy, respectively. It should be pointed out that even these
are only rough bounds. One reason for this is that the magnetic entropy is only about
10% of the total Sr 3 CuPtO.5 Iro.50 6 or Sr 3ZnPtO6 entropies, so the determination of
magnetic entropy involves the subtraction of two large numbers. Due to the sparseness of the data set and use of the trapezoidal rule, it is not unreasonable to suppose
errors in the integrals of at least a few percent. Another reason is that 50K is not a
very high temperature compared to J/kB, so the applicability of this method is not
clear. Finally, Ramirez made measurements on a different Sr 3 CuPtO.5 1ro. 5 0 6 sample
than was measured here. With these qualifications on the accuracy of this calculation
in mind, I conclude that the calculation is enough evidence to cast doubt on the possibility that a significant amount of the spin entropy of Sr 3 CuPtO.5 Iro.0 O 6 is removed
at temperatures where specific heat has not already been measured.
However, assume for a moment that the correct value for magnetic entropy re-
moved between 2.5K and 50K is on the low end of the range given, so that a good
deal of the spin entropy remains unaccounted for. In this case, and in light of Sigrist's
glass model for Sr 3 CuPt 0 .5 Iro.5O 6 below 1.7K (see Section 2.2), it is possible that some
of the spin entropy is "frozen in" below 1.7K, and this is the reason that the amount
of entropy removed below 1K is so small. If this hypothesis were correct, hysteresis
might be observed in the field-cooled versus zero-field-cooled heat capacity. However,
some of the heat capacity data points at 5kG were taken after cooling from 400mK
in a 7kG field, while others were taken after zero-field cooling. The points from the
two data sets showed no systematic differences. Similarly, some of the 10kG points
were taken after field-cooling at 1kG, while others were taken after zero-field cooling.
Again, both sets of points fall on the same curve.
The suppression of -/T by magnetic field confirms that the rise in U/T is associated with the spin degrees of freedom in Sr 3 CuPtO.5 1ro. 5 0 6 . Also, the fact that
the temperature dependence changes from roughly T 1/ 2 at 0 kG to T 3/ 2 at 10kG is
interesting. The T 3 / 2 dependence is expected for excitations with dispersion w =
Ak 2 , such as occur for spin waves in a three-dimensional ferromagnet. However, in
Sr 3 CuPtO.5 1ro.5 0 6 the T 3 / 2 dependence occurs only in a 10 kG field. For ferromagnetic
spin waves in a field w = poH + Ak 2 [60], leading to gap behavior in specific heat for
thermal energies below poH (po = (g/2)pIB). If po corresponds to one electron spin,
then gap behavior should be observed below 1K for a 10kG field. If /-o corresponds
to a cluster involving many electron spins, then gap behavior should be observed at
118
even higher temperatures. As it is, the data for Sr 3 CuPtO.5 Iro.0 O 6 follow T 3/2 below
0.5K. If there were even a small gap in Sr 3 CuPtO.5 Iro.0 O 6 , the fit to a = AT3/ 2 + B
would lead to a non-zero B. As it is, B is zero (to within experimental error), indicating that there is no evidence for the existence of a gap in Sr 3 CuPtO.5 Iro5 O 6 specific
heat at 10 kG. This suggests that Sr 3CuPt 5 IrO. 5 0 6 is not a simple three-dimensional
ferromagnet at these temperatures and fields. However, it does suggest that some
spin-wave-like excitations, propagating in three dimensions, are present. This in turn
suggests that interactions between spins on different chains may be important at these
temperatures and fields. This would not be too surprising, given the interpretation
of Beauchamp's data by Sigrist in terms of a glassy state involving interactions between clusters of spins on different chains (see Section 2.2), and given the possible
importance of interchain interactions in isostructural Ca 3 Co 2 0 6 and Sr 3 CuIrO6 (see
Chapter 1).
7.3
Discussion II: Comparison with RQSC Theory
There are two similarities between RQSC theory and the specific heat data. First,
there does appear to be some spin entropy still present in Sr 3 CuPtO.5 IrO. 5 0 6 below 1K.
Second, RQSC theory predicts 6 = 0.44 ± 0.02 in the universal regime. The data
are fairly well described between 0.1K and 0.4K by the functional form predicted by
theory, with 6 = 0.50 ± 0.07.
However, there are two important differences. First, the QMC results of Frischmuth and Sigrist predict 25 % of the total entropy lies below temperature T* = 0.03JO,
while Furusaki et al. predict that 24 % of the total entropy is missing from their hightemperature expansion results, valid to T 0.1Jo. This discrepancy between my data
and theory could be explained if only some fraction of the sample actually exhibits
RQSC behavior, say 10%. The fact that Beauchamp observes behavior quite different
from what is expected for RQSC supports this hypothesis. Another possible reason
for the discrepancy is that there may be anisotropy in the FM bonds, as suggested by
the high-field M versus H data on Sr 3 CuIrO6 at 5K (see Section 1.2). As a worst-case
scenario, the missing entropy can be recalculated
with (SFM) = 0- In this case, the missing entropy
than the observed value.
Second, theory did not anticipate that universal
be seen at such a high temperature. Frischmuth
(using the results of Section 2.1)
is 8%, still a factor of four larger
or nearly-universal behavior could
and Sigrist [22] suggest that the
scaling regime of Sr 3 CuPtO.5 1ro. 5 0 6 occurs only at temperatures well below JO/1000,
which would be roughly 40 mK here. Westerberg et al. [21] also question whether it
would be experimentally possible to observe scaling in specific heat. These conclusions
are based on the observation that certain choices for the initial distribution of coupling
constants and spins converge to the fixed point at higher temperatures than others.
The distribution that seems to describe Sr 3 CuPtO.5 1r o.5 0 6 , namely delta functions at
JO and -JO, converges very slowly. The measured field dependence could give insight
into the actual distribution of J in this temperature range, which could be useful in
addressing this discrepancy.
119
7.4
Discussion III: Comparison with Beauchamp
Results
Perhaps the strongest evidence against the applicability of RQSC theory to the measured Sr 3 CuPt*5 Ir0 .5O6 heat capacity data is the AC susceptibility data of Beau-
champ. While those data do not seem to be a signature of a transition to a threedimensional, fully ordered ground state (see Section 2.2), they do indicate some kind
of qualitative change in the magnetic behavior of the material. As discussed in Section 6.5, qualitative AC heat capacity data above 1 K suggest the possible presence
of a small peak in heat capacity centered at 1.5 K. The width is typical of a threedimensional ordering transition, ±10 %T, [2]. The amount of entropy contained in the
peak cannot be calculated directly from the data, since it was taken off-plateau. However, over this narrow range one would assume that A is temperature-independent, so
the data will equal the actual heat capacity times a scaling factor greater than one.
Hence integration of this data will place an upper limit on the entropy associated with
the peak. Again using the simple trapezoidal rule, the maximum entropy associated
with the peak is 0.01 J/mol/K, or 0.1% of 1.5R In 2. The amount of entropy found
in three-dimensional ordering transitions of systems that exhibit one-dimensional behavior at higher temperatures is expected to be a fairly small fraction of the total
spin entropy. As an extreme example, only 1% of the total spin entropy is removed in
the ordering transition of TMMC [2], with the rest lost to short-range order at higher
temperatures. Other materials lose around 10% in the ordering transition. Hence, if
the feature in Sr 3 CuPtO.5 ro. 5 0 6 is real, it seems unlikely that it is associated with a
transition to a three-dimensional, fully ordered ground state.
Are there any models that would be consistent with the susceptibility and specific
heat data? Recall the model proposed by Sigrist (Section 2.2), who postulated that
the distribution of bonds in the Sr 3 CuPt. 5 Ir 0 .5 O 6 chains is not truly random. This
led to large FM "clusters" at low temperatures, which then correlated to form a glassy
state at 1.7K, where the peak in X is observed. The absence of a pronounced feature
in specific heat around 1.7K also supports this idea of a transition to a glassy state.
On the other hand, the formation of such a state is typically evidenced in specific heat
as a broad maximum 20% above the "freezing temperature" Tf (Tf = 1.7K here),
and a nearly linear dependence of specific heat on T well below Tf [61]. Pursuing
this, I found that the low-temperature specific heat (below 0.4K) could be fit to a-=
aT+3To2 , with x2 = 4.4, a = 0.060 ± 0.009 J/mol/K 2 , = 0.068 t 0.003 J/mol/K 2 .
The To2 term is consistent with a random exchange Heisenberg AF chain model [62],
but I see no clear physical motivation for such a term in Sr 3 CuPtO. 5 ro.5 0 6 . Moreover,
there is no evidence for a broad maximum in the AC data above 1.7K, although a
broad feature may be difficult to detect given that some of the change in the AC data
with temperature is due to changes in D(w, T).
Perhaps a consistent model for the susceptibility and specific heat data is that
there are some parts of the sample in which the bonds are distributed randomly, and
other parts in which they are not. Hence it could be that part of the sample shows
RQSC or RQSC-like behavior, which is detected in the heat capacity, while the rest
120
contains long FM chain sections and forms a spin glass state at 1.7K. Given the M
versus H data shown in Figure 2-8, I estimate that the saturation M will not be much
higher than 100 emu/mol. Furthermore, if all spins in FM segments are aligned, and
if a powder sample for which only } of the spins align with the field is assumed (as is
the case for Sr 3 CuIrO6 ), a saturation M of roughly 3000 emu/mol would be expected
for a powder sample. Hence it seems that these long FM chain sections make up
only a small part (a few percent) of the sample, whereas they would have to make
up 90% of the sample if the measured entropy is to be reconciled with the "missing
entropy" predicted by theory (see Section 7.3). Therefore it seems that this model is
not consistent with the two data sets.
Other susceptibility measurements will help to clarify the situation further, as
would better heat capacity measurements above 1K. In particular, for a spin glass
Tf generally shows a dependence on the frequency at which AC susceptibility is measured [34]. Also, the static (zero frequency) susceptibility should show characteristic
hysteresis behavior when the sample is cooled in zero field versus in a finite field.
7.5
Discussion IV: Miscellaneous Interpretations
One other possible explanation for the data is that what is observed on the
Sr 3 CuPt. 5 Ir0 .5O 6 calorimeter is not due to the Sr 3 CuPtO.5 Iro. 5 O 6sample. This is
highly unlikely. The empty and Sr 3 CuPtO.5 Iro. 5 0 6 calorimeters are constructed to
be identical with the exception of the presence or absence of Sr 3 CuPtO.5 1r o.5 0 6 , and
the heat capacities of both calorimeters were measured during the same cooldown.
For the relaxation data, empty calorimeter data was taken just before or after Sr 3 CuPt. 5 Ir. 5 O 6 calorimeter data at a given temperature.
The quantitative.
agreement between the AC and relaxation results at zero field shows that the
Sr 3 CuPtO.5 1r o .5 0 6 results can not be explained by some problem with one of the
measurement techniques. The calorimeter thermal parameters given by the transfer functions and power/temperature curves suggest that the thermal properties of
the calorimeters are suitable for reliable heat capacity measurement. Also, the field
dependence of empty and Sr 3 CuPtO.5 Iro. 5 O 6 calorimeter heat capacities are differ-
ent, suggesting a spin contribution to the heat capacity that must be associated
with the Sr 3 CuPtO.5 1r o .5 0 6 sample. Finally, the fact that the two calorimeters have
comparable heat capacities at the highest fields indicates that magnetism in the
Sr 3 CuPtO.5 1ro.0
6
sample is the most likely source of any differences between them at
lower fields.
Another possible explanation is that the low-temperature rise in ds/dT is due to
Sr 3 CuPtO.5 Iro.0 O6 , but that some physics other than RQSC is responsible for it. For
example, the rise could be due to a Schottky anomaly. The latter could in principle
be caused by some crystal field splitting in the environment of the copper or iridium
ions, or by nuclear hyperfine splitting (HFS) in the copper (nuclear spin-!) or iridium
(nuclear spin 1). The crystal field produces a term DS2 in the Hamiltonian, and hence
produces no splitting for spin-- ions such as copper and iridium in Sr 3 CuPt. 5 Ir 0.5 O 6 .
The nuclear hyperfine contribution is more worrisome, since the strength of such
121
interactions are typically 10 - 100 mK [63]. However, C(T) oc T- for the Schottky
anomaly at temperatures well above the peak [63]. Since this dependence clearly
does not describe the zero field data, HFS can not explain the low-temperature rise
in ds/dT.
To summarize this discussion, my specific heat data show that there is still spin
entropy present in Sr 3 CuPtO.5 Iro.50
6
below 1K. Furthermore, the zero-field data be-
low 0.4K fits reasonably well to the scaling law predicted by RQSC theory. However,
there are many inconsistencies between the behavior of Sr 3 CuPtO.ro.5 0 6 and the the-
ory. First, the theory does not predict the scaling law to be observed in specific heat
at such a high temperature, given the initial bond distribution for Sr 3 CuPtO. 5 Iro.50 6 .
Second, even if the scaling law continues to T = 0, the amount of entropy accounted
for by the data is much less than predicted by the theory. Moreover, given the data of
Ramirez at higher temperatures, it is at least conceivable that most or even all of the
spin entropy is already accounted for at temperatures above 0.1K. The specific heat
data show no transition to a long-range ordered state between 0.1K and 2K at zero
field, and at 10kG show behavior characteristic of spin-waves in a three-dimensional
FM. The AC susceptibility results of Beauchamp show that there is physics other
than RQSC, perhaps some sort of spin-glass transition involving interactions between
different chains, present in Sr 3 CuPtO.5 1ro*5 0 6 as well. The possible presence of three-
dimensional spin waves at 10kG, in combination with the possible glass transition
indicated by the AC susceptibility at 1.7K, and the importance of interchain interactions in isostructural Ca 3Co 2 0 6 and possibly Sr 3 CuIrO 6 (see Chapter 1), all suggest
that a full understanding of the physics of Sr 3CuPtO.5 Iro. 5 0 6 may require consideration
of interactions between spins on different chains.
7.6
Prospects for Adiabatic Demagnetization of
Sr 3 CuPt 0 .5 Ir 0 5 O
6
As mentioned in Chapter 1, interest in Sr 3 CuPtO.5 1ro.5 0 6 has been motivated in part
by the possibility, suggested by the theory, that the material may be useful for adiabatic demagnetization. The specific heat data shows that the ordering temperature
for Sr 3 CuPtO. 5 IrO. 5 0 6 is below 100 mK, and that there is spin entropy below 1 K
that can be suppressed by application of relatively low (10 kG) fields. Hence it is
worthwhile to compare Sr 3 CuPtO.5 Iro.5
with paramagnetic salts traditionally used
for adiabatic demagnetization, to see if Sr 3 CuPtO.5 Iro.0 O6 might offer some advantages
over these other materials.
There are several materials properties that are desirable for adiabatic demagnetization [42]. First, it should be possible to remove a significant fraction of the spin
entropy with a modest field, and to do so starting at a temperature of 1 K. In this
case, a pumped 4 He cryostat will provide sufficient precooling, and the required field
can be obtained with an electromagnet or even a (movable) permanent magnet. Second, the material should have an ordering temperature T, below the temperature
range of interest. Third, the material should have a large zero-field specific heat at
122
the low end of the temperature range of interest, so that it will warm up slowly after
demagnetization.
Traditionally, the first criterion is tested with measurements of (OM/OT)H. Specific heat gives no insight into this thermodynamic quantity; however, the fact that
C/T rises with decreasing temperature indicates that there is some spin entropy
available at 1 K. As for the second criterion, T, is below 100 mK, so the material
could be used at least down to this temperature. If the dominant interaction between the chains is dipolar, then as discussed in Section 2.1 an ordering temperature
of 1 mK is conceivable. This would be competitive with cerium magnesium nitrate
(CMN), a well-studied material with T, ~ 2 mK. Finally, in Figure 7-7 the zerofield specific heats of several common paramagnetic salts is compared with that of
Sr 3 CuPt. 5 Ir0 .5 O 6 . Apparently Sr 3CuPt. 5 Iro.5 O 6 is inferior to these materials in this
respect.
10
0
0
0
0
0
0
0
0
0
E
0
0.1
CL
C-)
0.01
-
Theory
a CPA
UM
no.
0.001
0. 01
0.02
0.1
0.05
0.2
0.5
1
Temperature [K]
Comparison of various paramagnetic salt specific heats with
Figure 7-7:
Sr 3 CuPt0 .5 Iro. 5O6 . The dotted line is an extrapolation of the power law predicted
by theory for CPI in the universal regime.
These considerations suggest that Sr 3 CuPtO.5 1ro. 5 0 6 does not provide any sig-
nificant advantage over traditional materials used for adiabatic demagnetization to
1 mK. Of course, these materials have not been in general use for refrigeration for
more than two decades, since dilution refrigerators-which offer access to the same
temperature range and provide continuous refrigeration-have become commercially
available. However, the temperature range below 1 mK is currently only accessible
via nuclear demagnetization of copper or PrNi5 . If Sr 3 CuPtO.5 1ro.5 0 6 has T, < 1 mK,
123
and obeys RQSC theory to those temperatures, it could be quite valuable as a refrigerant. The main advantage it would have over nuclear demagnetization is that
the electron spins, rather than the nuclear spins, provide the cooling. The electron spins are thermally well-connected to the phonons, while the nuclear spins are
only weakly coupled (through the electron spins) [18]. Hence the time required
to cool the Sr 3 CuPt. 5 Iro.5O 6
lattice after demagnetization would be dramatically
shorter than that required to cool the lattice with nuclear demagnetization. Also,
the fields required for nuclear demagnetization are large, typically many tesla. For
Sr 3 CuPt. 5 Ir 0 .5O6 , the behavior of C at 130 mK suggests that much lower fields would
be required, 10 kG or less. (This difference in field scale is due to the relative moment of nuclear and electron spins). One severe disadvantage of Sr 3 CuPtO.5 1ro. 5 0 6 for
nuclear demagnetization, as compared with metals such as copper or PrNi5 , is its low
thermal conductivity.
The best way to test whether Sr 3CuPt. 5 Ir. 5 O6 could compete with nuclear de-
magnetization would be to do the adiabatic demagnetization experiment directly.
This will be discussed further in Section 7.7 below.
7.7
Future Work
It is apparent from the data taken in this thesis that the basic design principles
for the calorimeter are sound, and that it is possible to measure specific heat on
100 mK < T < 1.0 K and in fields to 70 kG. However, a couple of small changes
would improve the calorimeter further. First, the copper shim thermal link fingers
and link wire should be removed and replaced with high-purity silver. A single silver
wire could be wound on the surface of the plate in the same way that the PtW
heater was, and the same wire used for the thermal link to the bath. Second, it
would be desirable to further sharpen and narrow the vespel needles, as these may
contribute to the two-exponential behavior seen in the empty calorimeter at the lowest
temperatures. Also, a peg arrangement that would be symmetric with respect to top
and bottom plates would be desirable, as the asymmetry in the current design may
be responsible for the asymmetry seen in empty calorimeter heat capacity measured
with different heater/thermometer combinations.
Since the AC method is more precise than the relaxation method, and since data
acquisition and analysis is easier, it would also be desirable for AC calorimetry to be
the primary measurement method, with relaxation calorimetry used only as a check
on the AC results. At present, the reverse is true. I have shown that it is possible to
obtain accurate heat capacity data with the AC method, and future measurements
should be approached with the goal of establishing the AC method as primary. This
would have the additional advantage that, to my knowledge, there are no specific
heat experiments in operation today that are designed to measure below 1K and in
fields to 7T using the AC method.
As for Sr 3 CuPt 0 .5 Ir 0 .5 O 6 in particular, it would be desirable to do more careful,
field-dependent, measurements of heat capacity on 1.0K < T < 4.0 K. It is not
clear whether this apparatus is appropriate for such measurements-the addenda heat
124
capacity is larger than that of 105mg Sr 3 CuPtO.5 1r o .5 0 6 at those high temperatures
(see Figure 4-4). More AC measurements could be undertaken, in hopes of verifying
or refuting the presence of the small feature around 1.5 K that can not be ruled out
by the data taken thus far. Also, neutron diffraction below 2K in fields to 10kG could
give valuable insights into the nature of the spin ordering in the material.
From the perspective of RQSC theory, another important study would be the p
dependence of Sr 3 CuPt 1 _JIrpO 6 specific heat. This would be useful for comparison
with Beauchamp's data, and to see if the amount of missing entropy accounted for
below 1 K changes with p. Also, it is important to resolve the issue of anisotropy in
Sr 3 CuPtjp.IrO
6
, perhaps through a study of M versus H to high (20T) fields at low
(5K) and high (100K) temperatures.
From the perspective of adiabatic demagnetization, the data presented in this
thesis provide some motivation to proceed with a refrigeration experiment involving
Sr 3 CuPt* 5 Ir 0 .5 O 6 . Of course, it would be prudent to complete studies of p-dependent
specific heat and of anisotropy before proceeding with refrigeration. Initially, such an
experiment would be performed in a dilution refrigerator with precooling to 100 mK,
and would require a magnet capable of producing at least 10 kG fields. A large
quantity of Sr 3 CuPtO.5 Iro. 5 0 6 would be necessary, at least several grams. Chip resistors would no longer be adequate for thermometry-magnetic thermometry based on
CMN would be a relatively straightforward alternative for initial experiments. Also,
a superconducting heat switch would have to be developed. If cooling to 1 mK or
below were possible, careful consideration would have to be taken of heating due to
electrical noise, RF, and mechanical vibration if it were desired to actually observe
temperatures of 1 mK and below. Rough experiments on the facility here at MIT
may be able to confirm T, < 30 mK. Given the amount of development necessary for
even an initial experiment, and the limitations of the available apparatus, it may be
advisable to make use of an existing demagnetization refrigeration facility for such
experiments.
125
126
Appendix A
Calorimeter Conductance from
Power/Temperature Curves
I derive equations for the thermal conductance K, of the sample dough given the
slopes m of power/temperature curves measured with various heater/thermometer
combinations. There are two cases relevant to the data taken in this experiment: first,
for two curves measured with the same thermometer but different heaters; second,
for two curves measured with the same heater but different thermometers.
The thermal circuit for the first case is shown in Figure A-1. Power is applied
to either the top or bottom plate, and temperature always measured at the bottom
thermometer. The thermal circuit is simply analyzed by repeated application of
energy conservation and of Ohm's Law for a thermal circuit. The latter is
Q= KAT
= AT/R
(A.1)
(A.2)
where Q is the power through the thermal resistance, AT the temperature change
across the resistance, R the thermal resistance, and K its inverse, the thermal conductance. For power QLp applied to the top heater,
Qto =
TO
Rb
+
TO
"
Rb+ R(
(A.3)
RR
(A.4)
and
Tto - Tot =
sRb + Rs
(A.3) and (A.4) are solved for top in terms of Tbot and the thermal resistances.
Then the slope mbt of the power temperature curve is given by mbt = dQLp/dTot. In
this case,
mt
=
Rb
Rb+RS
(A.5)
R,+Rb
Here, I assume that the two thermal links have the same thermal resistance.
127
Rs
TOP
Tbot
Rb
Rb
Figure A-1: Thermal circuit for case 1. Temperature Tbot is measured with the bottom
calorimeter thermometer, referenced to the bath temperature. Power Q0t , is applied
to top heater or Qbot is applied to bottom heater.
128
Similarly, for power
bot
applied to the bottom thermometer, Ohm's Law yields
Tb
Qbot =T
+
Rb
Tb
(A.6)
T
Rb + R,
Hence
mTbb
+
Rb
(A.7)
R, + Rb
Combining (A.5) and (A.7) and solving for R8 ,
RS
1
t
mbt
(A.8)
- mbb
Mbb
This result was used to determine R, from mbt and mbb measurements on the
empty calorimeter.
Note that one can also solve these equations for Rb. The result is
Rb =
1
mbt
(A.9)
(1 + mbt)
mbb
Plugging in measured values for the empty calorimeter shows that Rb is larger
than 2/mbb or 2/mbt by 5% at 127 mK and by 1% at 400 mK. Given that the
approximation that the two thermal links have the same conductance may not be
good to better than 5%, I did not correct the empty calorimeter relaxation heat
capacity data for this effect.
Now I turn to the second case, two power/temperature curves measured with
the same heater but different thermometers. The situation is shown in Figure A2. Power is always applied to the bottom heater, and power/temperature curves
measured with either the bottom or top thermometer. For the bottom thermometer,
Ohm's Law yields
(A.10)
Q = KT + K,(Tb - T)
But also
K(Tb - Tt)= 1
1
1
(A.11)
Tb
KS+
Combining (A.10) and (A.11) and again noting mTbb
=
1
mnb=K+
K
do/dT
(A.12)
Similar analysis applies for power/temperature curve measurement with the top
thermometer. Here, (A.11) is used along with
Ks(Tb - T)
=
Tt =
129
KT
(A.13)
Ks Tb
Ks + K
(A.14)
Ks
TOP
Tbot
K
K
Figure A-2: Thermal circuit for case 2. Temperatures is measured with the bottom
(Tbot) or top (Top) calorimeter thermometers, referenced to the bath temperature.
Power Q is applied with the bottom heater.
130
Hence
(A.15)
K + K(K + K)
Ks
Combining (A.12) and (A.15) yields
mb
=
Ks = mntb [( mtb - mbb ) 2
mbb
K=
+ 2 mtb
mbb
1+ Mb
Mtb
- mbb_
mbb
(A.16)
(A.17)
These results were used to determine Ks from mtb and mbb measurements on the
Sr 3 CuPtO.5 IrO. 5 0 6 calorimeter. The K equation indicates that at 145 mK 2K is 2%
lower than 2 mtb, and at 306 mK 2K is less than 1% lower than 2 mtb. Again, no
correction was applied to the K data for this effect due to the fact that the two
thermal link wire thermal conductances may in fact differ by a few percent.
131
132
Appendix B
Exponential Fits with Instrumental
Response
In this portion of the Appendix, I compute the expected output signal AT(t)
from the SR830 setup used for relaxation heat capacity measurements in the
Sr 3 CuPtO.5 1r o .5 0
6
experiment.
For the one-exponential case, the input signal AT (t) is given by
ZAT (T)
T< 0
T> 0
A7T(O)
AT (0)e- /71
=
(B.1)
The normalized step response of the SR830 setup is
s (T) =0
T<O0
T>O0
~et/T
(B.2)
The impulse response h(r) is easily obtained from s(T) via h(T)
h(t) =
=
is(T) [64]:
(B.3)
T>r
Then AT(t) is obtained via the convolution
AT(t) =
JM ATi(T)h(t -
T)dT
(B.4)
Here, the integrand is non-zero only for T < t. Considering only the case t > 0, I
have
Jt AT(0) e-T/T e(t_)/r dT
(B.5)
AT(t) =
AT (0) e_(tT)
dT +
-- 00
0
r
Tr
The final result is
AT(t)
-
1
T7(0),
--
L
133
Tr
t/7] I
(B.6)
A similar calculation holds for the two-exponential case, where
A T)
{
T< 0
T/
AT()
Ae-7*1^+
Be--Tl'rB
(B.7)
T>0
Here,
AT(t)
A
etlA+
TA
B
etT-B
TB
_
[A
TITA
TA
-/r
+BTr IB
(B.8)
TB
Equations (B.6) and (B.8) were used to fit all A T(t) measured with the SR830
setup.
134
Appendix C
Schwall Model
The calorimeter model used by Schwall et al. is shown in Figure 3-1. The two lumps
are assumed to have different temperatures T and T2 in general, and the bath is
taken to be at zero temperature. The heat equations for this one-dimensional system
are
C1iT = -Ks(T 1 - T2 )
C 2 t 2 = Ks(T 1 - T2 )-KT
(C.1)
(C.2)
2
The initial condition, appropriate for the relaxation method, is T1 (0) = T2 (0) = Ti.
This system of equations can be solved for T (t) and T2 (t) using standard methods of
linear algebra. In the Schwall model, T2 (t) is recorded during a relaxation experiment.
Hence in the following all results are based on the T2 (t) solution.
Recall that T2 (t) satisfies
(C-3)
T 2 (t) = Ae-'I'A + Be-t'TB
The fit parameters A, B, TA, and TB are related to the thermal parameters of the
model by the equations
A
B
=
i
2
+ C1
1+
(C+Z
C1
+ -K
_F K
K
(C.4)
4 2
+
(1+
)2
(C.5)
~~
K/Ks
(C.6)
K
4I
2C 2
TB
K
K,
=Ti-A
202
TA
1
K
-2
K/KK
(C.7)
K
1+ +E)
1
135
+
K
0
C-
1C2KL2
Ks
C1
C2
K
K
Figure C-1: Thermal circuit for two-link Schwall model. All temperatures are referenced to the bath temperature, and T2 (t) is measured.
Since it was non-trivial to obtain Equations (8) in Schwall's paper from the solution for T2 (t), and since these equations led directly to the final result 3.4, I mention
here the tricks I used to obtain them. Equations (8) were
C
=
T2=
KbT1(1- KbT2/C
2)
- C 2±+ KT 2
BT
[TKb(A
+ B)2 - AC2]
(C.8)
(C.9)
The first equation can be obtained by examining 1 and - -+
) and eliminating
TB
TBTA
'rA
K, between them. Examination of A/B and eliminating
the square root terms in favor
of 1/TA and 1/TB leads to the second equation.
C1, C2, and Kf were extracted from the equations for the fit parameters by first
plugging the fit parameters into three equations. The first was the equation give
above for A.
The second was C1 + C2 = Cmeas, where Cmeas was the total heat
capacity measured using the Schwall model. The third was
Kf
=
0102
K TATB
(C.10)
which is found by combining the equations for TA and TB above. Mathematica was
used to extract C1,C2, and Kf, which were given implicitly by these three equations.
136
The Schwall model with two separate thermal links of equal conductance can be
solved similarly. The model is shown in Figure C-1. The heat equations here are
Citi
=
C 2t 2
(C.11)
(C.12)
-Ks(T 1 - T 2 ) - KT1
Ks(T 1 - T2 ) - KT2
Here, the initial condition is T1 (O) = Tio, T2 (0) = T20 . Note that T10 # T20,
although they become equal in the limit K/K, -+ 0. However, the solution for the
fit parameters is (again considering only T 2 (t))
A
=
A,
B
C1
_+
20
=
2
(1++
_
C)1
1
1
1()
_K I)2)]
')
(C.14)
T20 - A
K
TA
(1
C1
K.
) 1)-
2C,K (1+ 2)(1 +
_C
,C
KKC
2
[1+11 _
KB (i+2)(i+)
_
1±2)
(C.15)
_F]
2
11
Equations similar to Equations (8) of Schwall's paper can be obtained for this
model, using the same algebraic tricks mentioned above. The results are
TB
- TB) K,-TATBK
(2K)(TA
C
BC 2T2
KTA A+ B)
=
-
2
(C.1)
C1,
AC 2
These can be combined to yield
C 1 + C 2 = 2K
TA±BTB
A
K
A+B
ATB+BTA
A+B
K2
AB
K T+
C1
(C.19)
Since the standard Schwall model describes the relaxation data better than this
two-wire model (see Section 6.4.2), this equation was not used to analyze data. It is
presented here for completeness.
Finally, Mathematica was again used to determine C1, C 2 , and Kf given the fit
parameters. First C1 + C2 was determined using (C.19). Then this result for C1 + C 2 ,
along with the equation for A and the result (again found by combining the TA and
TB results)
K
1
2
CC 2
KACB
were used to obtain C1, C2, and Kf.
137
- K
(C.20)
138
Appendix D
Model for AC Transfer Function in
Presence of T2 Effect
In this appendix, I present a model consistent with that suggested in the text (Section 6.4) for calorimeters exhibiting T2 effect. Then I compute the expected transfer
function for such a model.
The model from the text was that the two quartz plates and sample dough formed
effectively a single thermal lump, and that there was a weak connection to some other
part of the calorimeter (copper fingers or vespel pegs) not between the calorimeter
heater and thermometer. The two thermal link wires were treated effectively as a
single wire of twice the conductivity connected to the calorimeter. This model led to
two-exponential AT(t) behavior according to the Schwall model. The thermal circuit
for this model is shown in Figure D-1. This differs from the Schwall model in that
the two lumps are treated as continuous slabs, with temperature varying over the
thickness of the slab. The transfer function was computed for this geometry, and
for the same geometry with heater and thermometer reversed. This reversal led to a
transfer function that did not improve the agreement between the measured function
and the standard two-wire model. On the other hand, the function calculated from
the exact geometry of Figure D-1 did improve the agreement; hence, this geometry
is more representative of the actual experimental situation.
The first matrix equation describing this situation applies to heat flow through
the slab representing the calorimeter (two quartz plates plus dough):
To
A B
C D
Tm
q.)
q
a
TO )
where
2 K(D.1)
A
B
=
=-
cosh 0
sinh 0
a
139
(D.2)
(D.3)
TM
C, K
TO
Kb
Figure D-1: Thermal circuit for calculation of Schwall model transfer function. The
heater applies flux j at position 0. The thermometer is located at position m.
140
KO
sinh 0
a
C==D
(D.4)
0
=cosh
(D.5)
a is the cross-sectional area of the slab, and the other parameters are as shown in
Figure D-1. The second equation, representing the weakly connected part of the
calorimeter (probably vespel pegs or copper fingers) is
(
F )(Tm
H
q;n'
where
wCw
E
=
F
=
(D.6)
V 2K,
cosh 6
(D.7)
sinh ,9
(D.8)
a
G
=
-
K
sinhO0
a
H =
(D.9)
(D.10)
cosh Ow
and the cross-sectional area a is assumed to be the same as for the calorimeter lump.
Solving the system for Tm, I find
Tm
=
B
-
-AD
A
Ib
+
G A-B
H C
D aa
-B
(D.11)
Ca
The measured transfer function was then fit to wTm. Note that as lump one disappears, i.e. Cw
0, Kw -+ 1, Tm reduces to Sullivan and Seidel's exact solution for
their geometry [38].
+
141
142
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